Koszul calculus of preprojective algebras

TL;DR

This paper investigates the Koszul calculus of preprojective algebras, revealing vanishing properties, dualities, and a generalized Calabi-Yau condition, especially for ADE Dynkin graphs, with explicit calculations and a duality theorem.

Contribution

It introduces the concept of Koszul complex Calabi-Yau (Kc-Calabi-Yau) algebras, generalizes Calabi-Yau properties, and provides explicit calculations for ADE Dynkin graphs.

Findings

Koszul calculus vanishes in degrees p>2 for certain preprojective algebras

Isomorphism between (co)homological calculus via degree exchange p and 2-p

Preprojective algebras of ADE Dynkin graphs are Kc-Calabi-Yau of dimension 2

Abstract

We show that the Koszul calculus of a preprojective algebra, whose graph is distinct from A$\_1$ and A$\_2$, vanishes in any (co)homological degree $p>2$. Moreover, its (higher) cohomological calculus is isomorphic as a bimodule to its (higher) homological calculus, by exchanging degrees $p$ and $2-p$, and we prove a generalised version of the 2-Calabi-Yau property. For the ADE Dynkin graphs, the preprojective algebras are not Koszul and they are not Calabi-Yau in the sense of Ginzburg's definition, but they satisfy our generalised Calabi-Yau property and we say that they are Koszul complex Calabi-Yau (Kc-Calabi-Yau) of dimension $2$. For Kc-Calabi-Yau (quadratic) algebras of any dimension, defined in terms of derived categories, we prove a Poincar\'e Van den Bergh duality theorem. We compute explicitly the Koszul calculus of preprojective algebras for the ADE Dynkin graphs.

Equations

$${\theta }_M(f)={\omega }_0\underset{K}{⌢}f.$$

$${\theta }_{M{\otimes }_{A}N}(f\underset{K}{⌣}g)={\theta }_M(f)\underset{K}{⌢}g=f\underset{K}{⌢}{\theta }_N(g)$$

$$b_K=-[{\mathfrak{e}}_A,-{]}_{\underset{K}{⌣}},b^K=-[{\mathfrak{e}}_A,-{]}_{\underset{K}{⌢}},$$

$$H({\theta }_M)={\overline{\omega }}_0\underset{K}{⌢}-:{\mathrm{HK}}^{\bullet }(A,M)\to {\mathrm{HK}}_{2-\bullet }(A,M),$$

$${\mathrm{HH}}^{p+6}(A)\cong {\mathrm{HH}}^p(A),p>0$$

$$c\underset{K}{⌢}-:{\mathrm{HK}}^{\bullet }(A)\to {\mathrm{HK}}_{n-\bullet }(A)$$

$$d(a{\otimes }_k{a}_1\dots a_p{\otimes }_k{a}^{\prime })$$

$$+(-1{)}^{p}a{\otimes }_k{a}_1\dots a_{p-1}{\otimes }_k{a}_p{a}^{\prime },$$

$${\mathrm{HH}}_{\bullet }(A,M)={\mathrm{Tor}}_{\bullet }^{A^e}(M,A)=H_{\bullet }(M{\otimes }_{A^e}B(A),M{\otimes }_{A^e}d),$$

$${\mathrm{HH}}^{\bullet }(A,M)={\mathrm{Ext}}_{A^e}^{\bullet }(A,M)=H^{\bullet }({\mathrm{Hom}}_{A^e}(B(A),M),{\mathrm{Hom}}_{A^e}(d,M)).$$

$$\begin{array}[]{crcl}&M\otimes_{A^{e}}(A\otimes_{k}E\otimes_{k}A)&\rightarrow&M\otimes_{k^{e}}E\\ &m\otimes_{A^{e}}(a\otimes_{k}x\otimes_{k}a^{\prime})&\mapsto&a^{\prime}ma\otimes_{k^{e}}x\\ \\ \text{ and }&\mathop{\rm Hom}\nolimits_{k^{e}}(E,M)&\rightarrow&\mathop{\rm Hom}\nolimits_{A^{e}}(A\otimes_{k}E\otimes_{k}A,M)\\ &f&\mapsto&(a\otimes_{k}x\otimes_{k}a^{\prime}\mapsto af(x)a^{\prime})\end{array}$$

$${\mathrm{HH}}_{\bullet }(A,M)\cong H_{\bullet }(M{\otimes }_{k^e}{A}^{{\otimes }_k\bullet },b^H),$$

$${\mathrm{HH}}^{\bullet }(A,M)\cong H^{\bullet }({\mathrm{Hom}}_{k^e}(A^{{\otimes }_k\bullet },M),b_H).$$

$$b_p^H(m{\otimes }_{k^e}(a_1\dots a_p))$$

$$+(-1{)}^p{a}_{p}m{\otimes }_{k^e}(a_1\dots a_{p-1}).$$

$$b_H^{p+1}(f)(a_1\dots a_{p+1})$$

$$-(-1{)}^p{a}_{1}f(a_2\dots a_{p+1}).$$

$$W_p=\bigcap _{i+2+j=p}{V}^{{\otimes }_{k}i}{\otimes }_{k}R{\otimes }_k{V}^{{\otimes }_{k}j}.$$

$$d(a{\otimes }_k{x}_1\dots x_p{\otimes }_k{a}^{\prime })=a{x}_1{\otimes }_k{x}_2\dots x_p{\otimes }_k{a}^{\prime }+(-1{)}^{p}a{\otimes }_k{x}_1\dots x_{p-1}{\otimes }_k{x}_p{a}^{\prime },$$

$$F_1=\underset{i,j\in {\mathcal{Q}}_0}{⨁}A{e}_j\otimes e_{j}E{e}_i\otimes e_{i}A.$$

$$F_2=\underset{i_1,i_2,i_3,i_4\in {\mathcal{Q}}_0}{⨁}A{e}_{i_1}\otimes e_{i_2}E{e}_{i_3}\otimes e_{i_4}A$$

$${\mathrm{HK}}_{\bullet }(A,M)=H_{\bullet }(M{\otimes }_{A^e}K(A))\text{ and }{\mathrm{HK}}^{\bullet }(A,M)=H^{\bullet }({\mathrm{Hom}}_{A^e}(K(A),M)).$$

$$\tilde{\chi }=M{\otimes }_{A^e}\chi :M{\otimes }_{A^e}K(A)\to M{\otimes }_{A^e}B(A),$$

$${\chi }^{\ast }={\mathrm{Hom}}_{A^e}(\chi ,M):{\mathrm{Hom}}_{A^e}(B(A),M)\to {\mathrm{Hom}}_{A^e}(K(A),M).$$

$${\mathrm{HK}}_{\bullet }(A,M)$$

$${\mathrm{HK}}^{\bullet }(A,M)$$

$$b_p^K(m{\otimes }_{k^e}{x}_1\dots x_p)$$

$$b_K^{p+1}(f)(x_1\dots x_{p+1})$$

$${\mathrm{HK}}_p(A,k)\cong k{\otimes }_{k^e}{W}_p\cong \underset{i\in {\mathcal{Q}}_0}{⨁}{e}_i{W}_p{e}_i,$$

$${\mathrm{HK}}^p(A,k)\cong {\mathrm{Hom}}_{k^e}(W_p,k)\cong \underset{i\in {\mathcal{Q}}_0}{⨁}\mathrm{Hom}(e_i{W}_p{e}_i,\mathbb{F}).$$

$$(f\underset{K}{⌣}g$$

$$f\underset{K}{⌢}z$$

$$z\underset{K}{⌢}f$$

$$(f\underset{K}{⌣}g)\underset{K}{⌣}h$$

Full text

1 Introduction

Preprojective algebras are quiver algebras with quadratic relations, that play an important role in the representation theory of quiver algebras [27, 17, 1, 10], with various applications [15, 16] and many developments [14, 26, 8, 29]. In [29], the reader will find an introduction to the various aspects of the preprojective algebras in representation theory, with an extended bibliography. In our paper, we are interested in some homological properties linked to Hochschild cohomology.

In the last two decades, the Hochschild cohomology of preprojective algebras, as well as some extra algebraic structures, have been computed in several steps, as follows.

1. Erdmann and Snashall [18, 19] determined the Hochschild cohomology and its cup-product in type A.

2. Crawley-Boevey, Etingof and Ginzburg [14] determined the Hochschild cohomology for all preprojective algebras of non-Dynkin type (which are Koszul in this case [33, 30, 10]).

3. In type DE and characteristic zero, Etingof and Eu [21] determined the Hochschild cohomology and Eu [22] the cup product. The cyclic homology was computed in type ADE in [21].

4. Assembling and completing the previous results in characteristic zero, Eu gave an explicit description of the Tamarkin-Tsygan calculus [39] of the preprojective algebras in type ADE, that is, the homology and the cohomology, the cup product, the contraction map and the Lie derivative, the Connes differential and the Gerstenhaber bracket [23].

5. Eu and Schedler extended the ADE results to the case where the base ring is $\mathbb{Z}$, and obtained the corresponding ADE results in any characteristic [24].

In [7], a Koszul calculus was associated with any quadratic algebra over a field, in order to produce new homological invariants for non-Koszul quadratic algebras. We begin this paper by extending the Koszul calculus to quadratic quiver algebras. We shall compute the Koszul calculus of the preprojective algebras whose graphs are Dynkin of type ADE (the preprojective algebras are then finite dimensional). Except for types A1 and A2, these quadratic quiver algebras are not Koszul [33, 30, 10], so that the Koszul calculus and the Hochschild calculus provide different information.

Before presenting our computations, we state and develop a Poincaré Van den Bergh duality theorem [41] for the Koszul homology/cohomology of any preprojective algebra whose graph is different from A1 and A2. This theorem is formulated as follows and constitutes the first main result of the present paper. The duality is precisely part (ii) in this theorem.

**Theorem **\theDf

Let $A$ be the preprojective algebra of a (non-labelled) connected graph $\Delta$ distinct from $\mathrm{A}_{1}$ and $\mathrm{A}_{2}$, over a field $\mathbb{F}$. Let $M$ be an $A$-bimodule.

1. \edefitit(i)

The Koszul bimodule complex $K(A)$ of $A$ has length 2. In particular, $\mathop{\rm HK}\nolimits^{p}(A,M)=\mathop{\rm HK}\nolimits_{p}(A,M)=0$ for any $p>2$. 2. \edefitit(ii)

The $\mathop{\rm HK}\nolimits^{\bullet}(A)$-bimodules $\mathop{\rm HK}\nolimits^{\bullet}(A,M)$ and $\mathop{\rm HK}\nolimits_{2-\bullet}(A,M)$ are isomorphic. 3. \edefitit(iii)

The $\mathop{\rm HK}\nolimits^{\bullet}_{hi}(A)$-bimodules $\mathop{\rm HK}\nolimits^{\bullet}_{hi}(A,M)$ and $\mathop{\rm HK}\nolimits^{hi}_{2-\bullet}(A,M)$ are isomorphic.

In this statement, following [7], $\mathop{\rm HK}\nolimits_{p}(A,M)$ and $\mathop{\rm HK}\nolimits^{p}(A,M)$ denote the Koszul homology and cohomology spaces with coefficients in $M$, while $\mathop{\rm HK}\nolimits^{hi}_{p}(A,M)$ and $\mathop{\rm HK}\nolimits^{p}_{hi}(A,M)$ denote the higher Koszul homology and cohomology spaces. When $M=A$, these notations are simplified into $\mathop{\rm HK}\nolimits_{p}(A)$, $\mathop{\rm HK}\nolimits^{p}(A)$, $\mathop{\rm HK}\nolimits^{hi}_{p}(A)$ and $\mathop{\rm HK}\nolimits^{p}_{hi}(A)$.

In the general setting [7], the Koszul calculus of a quadratic algebra $A$ consists of the graded associative algebra $\mathop{\rm HK}\nolimits^{\bullet}(A)$ endowed with the Koszul cup product and, for all $A$-bimodules $M$, of the graded $\mathop{\rm HK}\nolimits^{\bullet}(A)$-bimodules $\mathop{\rm HK}\nolimits^{\bullet}(A,M)$ and $\mathop{\rm HK}\nolimits_{\bullet}(A,M)$, with actions respectively defined by the Koszul cup and cap products. The higher Koszul calculus of $A$ is given by the analogous data, adding the subscript and superscript hi. Sometimes (as will be the case with our computations in ADE types), these calculi are restricted, meaning that the data is limited to $M=A$, so that the restricted Koszul calculus consists of the graded associative algebra $\mathop{\rm HK}\nolimits^{\bullet}(A)$ and of the graded $\mathop{\rm HK}\nolimits^{\bullet}(A)$-bimodule $\mathop{\rm HK}\nolimits_{\bullet}(A)$ – similarly for the higher version.

Using Theorem 1 for $\Delta$ Dynkin of type ADE, we shall deduce the (higher) homological restricted Koszul calculus from the computation of the (higher) cohomological restricted Koszul calculus.

Part (ii) in Theorem 1 comes from an explicit isomorphism from the complex $C_{1}$ of Koszul cochains with coefficients in $M$, whose $p$th cohomology is $\mathop{\rm HK}\nolimits^{p}(A,M)$, to the complex $C_{2}$ of Koszul chains with coefficients in $M$, whose $p$th homology is $\mathop{\rm HK}\nolimits_{2-p}(A,M)$, described as follows.

**Proposition **\theDf

Let $A$ be the preprojective algebra of a connected graph $\Delta$ distinct from $\mathrm{A}_{1}$ and $\mathrm{A}_{2}$. Let $M$ be an $A$-bimodule. The Koszul cup and cap products are denoted by $\underset{K}{\smile}$ and $\underset{K}{\frown}$. Define $\omega_{0}=\sum_{i}e_{i}\otimes\sigma_{i}$, where the sum runs over the vertices $i$ of $\Delta$ and, for each vertex $i$, $e_{i}$ is the idempotent and $\sigma_{i}$ is the quadratic relation in $A$ associated with $i$.

For each Koszul $p$-cochain $f$ with coefficients in $M$, we define the Koszul $(2-p)$-chain $\theta_{M}(f)$ with coefficients in $M$ by

[TABLE]

Then $\theta_{M}:C_{1}\rightarrow C_{2}$ is an isomorphism of complexes. Moreover, the equalities

[TABLE]

hold for any Koszul cochains $f$ and $g$ with coefficients in bimodules $M$ and $N$ respectively.

The proof of Proposition 1 relies on some manipulations of the defining formula of $\theta_{M}$ with fundamental formulas of Koszul calculus [7], using actions involving $\underset{K}{\smile}$ and $\underset{K}{\frown}$. The fundamental formulas of Koszul calculus express the differential $b_{K}$ of $C_{1}$ and the differential $b^{K}$ of $C_{2}$ respectively as a cup bracket and a cap bracket, namely

[TABLE]

where $\mathfrak{e}_{A}:V\rightarrow A$ is a fundamental Koszul 1-cocycle defined on the arrow space $V$ by $\mathfrak{e}_{A}(x)=x$ for all $x\in V$.

In order to extract a generalised version of the 2-Calabi-Yau property from our Poincaré Van den Bergh duality (Theorem 1) for quadratic algebras, we apply this theorem to the left $A^{e}$-module $M=A^{e}:=A\otimes A^{op}$ viewed as an $A$-bimodule. We show that the complex of Koszul chains with coefficients in the left $A^{e}$-module $A^{e}$ is naturally isomorphic, as a right $A^{e}$-module, to the Koszul bimodule complex $K(A)$. Using the fact that the homology of $K(A)$ is isomorphic to $A$ in degree 0, and to [math] in degree 1, we obtain a generalisation of the 2-Calabi-Yau property, formulated as follows.

**Theorem **\theDf

Let $A$ be the preprojective algebra of a connected graph $\Delta$ distinct from $\mathrm{A}_{1}$ and $\mathrm{A}_{2}$, over a field $\mathbb{F}$. Let us denote by $K(A)$ the Koszul bimodule complex of $A$. Then the $A$-bimodule $\mathop{\rm HK}\nolimits^{p}(A,A^{e})$ is isomorphic to the $A$-bimodule $H_{2-p}(K(A))$ for $0\leqslant p\leqslant 2$. In particular, we have the following.

1. \edefitit(i)

The $A$-bimodule $\mathop{\rm HK}\nolimits^{2}(A,A^{e})$ is isomorphic to the $A$-bimodule $A$. 2. \edefitit(ii)

$\mathop{\rm HK}\nolimits^{1}(A,A^{e})=0$. 3. \edefitit(iii)

The $A$-bimodule $\mathop{\rm HK}\nolimits^{0}(A,A^{e})$ is isomorphic to the $A$-bimodule $H_{2}(K(A))$, which is always non-zero when $\Delta$ is Dynkin of type $\mathrm{ADE}$.

We then say that the preprojective algebra $A$ is a Koszul complex Calabi-Yau algebra of dimension 2. We generalise this definition to any quadratic algebra and any dimension $n$ in Definition 1 below, better formulated in terms of derived categories. Since there is an $\mathbb{F}$-linear isomorphism

[TABLE]

we say that the class $\overline{\omega}_{0}\in\mathop{\rm HK}\nolimits_{2}(A)$ is the fundamental class of the Koszul complex Calabi-Yau algebra $A$, by analogy with Poincaré’s duality in singular homology/cohomology [31]. In Definition 1, we give a stronger version of Definition 1 in order to obtain a Poincaré-like duality, that is, a duality isomorphism expressed as a cap action by a suitably defined fundamental class.

Let us remark that the $A$-bimodule structures in Theorem 1 are compatible with the Koszul cup and cap actions of $\mathop{\rm HK}\nolimits^{\bullet}(A)$ on $\mathop{\rm HK}\nolimits^{\bullet}(A,A^{e})$ and $H(K(A))$. These actions can be viewed as graded actions of left $\mathop{\rm HK}\nolimits^{\bullet}(A)^{e}$-modules, while the $A$-bimodules can be viewed as compatible right $A^{e}$-modules. So the isomorphism $\mathop{\rm HK}\nolimits^{\bullet}(A,A^{e})\cong H_{2-\bullet}(K(A))$ in Theorem 1 is an isomorphism of graded $\mathop{\rm HK}\nolimits^{\bullet}(A)^{e}$-$A^{e}$-bimodules. This enriched isomorphism is the expression of the stronger version of the Koszul complex Calabi-Yau property, as we shall see in Definition 1.

Note that if $\Delta$ is not Dynkin ADE, then $A$ is Koszul, so Theorem 1 enables us to recover the well-known result that $A$ is 2-Calabi-Yau in the sense of Ginzburg [14, 9]. However, if $\Delta$ is Dynkin ADE, then $A$ is not homologically smooth since its minimal projective resolution has infinite length, so that Ginzburg’s definition of Calabi-Yau algebras cannot be applied in this case [28]. Moreover, the restricted Hochschild calculus is drastically different from the restricted Koszul calculus, because by [20] there is a cohomological Hochschild periodicity

[TABLE]

and, consequently, there are non-zero spaces $\mathop{\rm HH}\nolimits^{p}(A)$ for infinitely many values of $p$. Even taking into account this 6-periodicity, the list of cohomological Koszul invariants consists only of $\mathop{\rm HK}\nolimits^{0}(A)$, $\mathop{\rm HK}\nolimits^{1}(A)$ and $\mathop{\rm HK}\nolimits^{2}(A)$ and is therefore shorter than the list of Hochschild invariants.

In [24], Eu and Schedler define periodic Calabi-Yau Frobenius algebras, for finite dimensional algebras only. Their main example is given by the preprojective algebras of Dynkin ADE graphs [24, Example 2.3.10]. Then the above cohomological Hochschild periodicity is a part of remarkable isomorphims in Hochschild calculus for any periodic Calabi-Yau Frobenius algebra [24, Theorem 2.3.27 and Theorem 2.3.47].

From Theorem 1, we are led to introduce a general definition.

{Df}

Let $\mathcal{Q}=(\mathcal{Q}_{0},\mathcal{Q}_{1})$ be a finite quiver, and let $\mathbb{F}$ be a field. Let $A$ be an $\mathbb{F}$-algebra defined on the path algebra $\mathbb{F}\mathcal{Q}$ of $\mathcal{Q}$ by homogeneous quadratic relations. Define the ring $k=\mathbb{F}\mathcal{Q}_{0}$, so that $A$ is regarded as a quadratic $k$-algebra. We say that $A$ is Koszul complex Calabi-Yau (Kc-Calabi-Yau) of dimension $n$, for an integer $n\geqslant 0$, if

1. \edefnit(i)

the bimodule Koszul complex $K(A)$ of $A$ has length $n$, and 2. \edefnit(ii)

$\mathop{\rm RHom}\nolimits_{A^{e}}(K(A),A^{e})\cong K(A)[-n]$ in the bounded derived category of $A$-bimodules.

In our context (that of quadratic algebras), Definition 1 is a definition of a new Calabi-Yau property, valid whether $A$ is finite dimensional or not. In this definition, we do not impose that $K(A)$ be a resolution of $A$, that is, $A$ is not necessarily Koszul, meaning that the bimodules $\mathop{\rm HK}\nolimits^{p}(A,A^{e})$ for $0\leqslant p\leqslant n-2$ may be non-zero. Under the assumptions of Definition 1, we verify that, if $A$ is Koszul, Definition 1 is equivalent to Ginzburg’s definition of $n$-Calabi-Yau algebras [28, 43]. We then prove a new Poincaré Van den Bergh duality for Kc-Calabi-Yau algebras, adapted to Koszul (co)homologies.

**Theorem **\theDf

Let $A$ be a Kc-Calabi-Yau algebra of dimension $n$. Then for any $A$-bimodule $M$, the $\mathbb{F}$-vector spaces $\mathop{\rm HK}\nolimits^{p}(A,M)$ and $\mathop{\rm HK}\nolimits_{n-p}(A,M)$ are isomorphic.

{Df}

Let $A$ be a Kc-Calabi-Yau algebra of dimension $n$. The image $c\in\mathop{\rm HK}\nolimits_{n}(A)$ of the unit $1$ of the algebra $A$ under the isomorphism $\mathop{\rm HK}\nolimits^{0}(A)\cong\mathop{\rm HK}\nolimits_{n}(A)$ in Theorem 1 is called the fundamental class of the Kc-Calabi-Yau algebra $A$.

In order to describe the duality isomorphism of Theorem 1 explicitly as a cap-product by the fundamental class for strong Kc-Calabi-Yau algebras, we shall use derived categories in the general context of DG algebras, as presented and detailed in the preprint book by Yekutieli [45]. Let us present briefly what we need in this general context.

We introduce the DG algebra $\tilde{A}=\mathop{\rm Hom}\nolimits_{A^{e}}(K(A),A)$. The complexes $K(A)$ and $\mathop{\rm Hom}\nolimits_{A^{e}}(K(A),A^{e})$ of $A$-bimodules have an enriched structure since they can be viewed as DG $\tilde{A}$-bimodules in the abelian category $A\text{-{Bimod}}$ of $A$-bimodules, in the sense of [45].

Denote by $\mathcal{C}(\tilde{A},A\text{-{Bimod}})$ the category of DG $\tilde{A}$-bimodules in $A\text{-{Bimod}}$ [45]. Let $M$ be an $A$-bimodule. For any chain DG $\tilde{A}$-bimodule $C$ in $A\text{-{Bimod}}$, $\mathop{\rm Hom}\nolimits_{A^{e}}(C,M)$ is a cochain DG $\tilde{A}$-bimodule in the abelian category VectF of $\mathbb{F}$-vector spaces (in $A\text{-{Bimod}}$ when $M=A^{e}$). For any cochain DG $\tilde{A}$-bimodule $C^{\prime}$ in $A\text{-{Bimod}}$, $M\otimes_{A^{e}}C^{\prime}$ is a cochain DG $\tilde{A}$-bimodule in VectF. The bounded derived categories $\mathcal{D}^{b}(\tilde{A},A\text{-{Bimod}})$ and \mathcal{D}^{b}(\tilde{A},\text{{Vect}{}{\mathbb{F}}}) are defined in [45]. However we do not know if the functors $\mathop{\rm Hom}\nolimits_{A^{e}}(-,M)$ and $M\otimes_{A^{e}}-$ from $\mathcal{C}^{b}(\tilde{A},A\text{-{Bimod}})$ to \mathcal{C}^{b}(\tilde{A},\text{{Vect}{}{\mathbb{F}}}) are derivable.

{Df}

Let $A$ be a Kc-Calabi-Yau algebra of dimension $n$. Then $A$ is said to be strong Kc-Calabi-Yau if the derived functor of the endofunctor $\mathop{\rm Hom}\nolimits_{A^{e}}(-,A^{e})$ of $\mathcal{C}^{b}(\tilde{A},A\text{-{Bimod}})$ exists and if $\mathop{\rm RHom}\nolimits_{A^{e}}(K(A),A^{e})\cong K(A)[-n]$ in the bounded derived category $\mathcal{D}^{b}(\tilde{A},A\text{-{Bimod}})$.

**Theorem **\theDf

Let $A$ be a Kc-Calabi-Yau algebra of dimension $n$ and let $c$ be its fundamental class. We assume that $A$ is strong Kc-Calabi-Yau and that the derived functors of the functors $\mathop{\rm Hom}\nolimits_{A^{e}}(-,A)$ and $A\otimes_{A^{e}}-$ from $\mathcal{C}^{b}(\tilde{A},A\text{-{Bimod}})$ to \mathcal{C}^{b}(\tilde{A},\text{{Vect}{}_{\mathbb{F}}}) exist. Then

[TABLE]

is an isomorphism of $\mathop{\rm HK}\nolimits^{\bullet}(A)$-bimodules, inducing an isomorphism of $\mathop{\rm HK}\nolimits^{\bullet}_{hi}(A)$-bimodules from $\mathop{\rm HK}\nolimits^{\bullet}_{hi}(A)$ to $\mathop{\rm HK}\nolimits^{hi}_{n-\bullet}(A)$. For all $\alpha\in\mathop{\rm HK}\nolimits^{p}(A)$, we have $c\underset{K}{\frown}\alpha=(-1)^{np}\alpha\underset{K}{\frown}c$.

Let us describe the contents of the paper. In Section 2, we extend the general formalism – including some results – of Koszul calculus [7] to quadratic quiver algebras. In Section 3, we introduce a right action which is an important tool in order to adapt the definition of Calabi-Yau algebras to quadratic quiver algebras endowed with the Koszul calculus instead of the Hochschild calculus. The Poincaré Van den Bergh duality for preprojective algebras is presented in Section 4, where Theorem 1, Proposition 1 and Theorem 1 of our introduction are proved. In Section 5, we define our generalisations of Calabi-Yau algebras and we thoroughly explain the new objects and remaining results outlined in the introduction. Section 6 is devoted to the computations of the Koszul calculus in ADE Dynkin types. As an application of the computations, we prove that the spaces $\mathop{\rm HK}\nolimits^{0}_{hi}(A)$, $\mathop{\rm HK}\nolimits^{1}_{hi}(A)$ and $\mathop{\rm HK}\nolimits^{2}_{hi}(A)$ form a minimal complete list of cohomological invariants for the ADE preprojective algebras.

Acknowledgement.

The authors are grateful to the anonymous referee, whose useful and detailed comments helped to improve this manuscript.

2 Koszul calculus for quiver algebras with quadratic relations

2.1 Setup

Let $\mathcal{Q}$ be a finite quiver, meaning that the vertex set $\mathcal{Q}_{0}$ and the arrow set $\mathcal{Q}_{1}$ are finite. Let $\mathbb{F}$ be a field. The vertex space $k=\mathbb{F}\mathcal{Q}_{0}$ becomes a commutative ring by associating with ${\mathcal{Q}}_{0}$ a complete set of orthogonal idempotents $\left\{e_{i}\,;\,i\in{\mathcal{Q}}_{0}\right\}$. The ring $k$ is isomorphic to $\mathbb{F}^{|\mathcal{Q}_{0}|}$, where $|\mathcal{Q}_{0}|$ is the cardinal of $\mathcal{Q}_{0}$. Throughout the paper, the case $|\mathcal{Q}_{0}|=1$ will be called the one vertex case, which is equivalent to saying that $k$ is a field. Koszul calculus over a field $k$ is treated in [7].

For each arrow $\alpha\in\mathcal{Q}_{1}$, denote its source vertex by $\mathfrak{s}(\alpha)$ and its target vertex by $\mathfrak{t}(\alpha)$. The arrow space $V=\mathbb{F}\mathcal{Q}_{1}$ is a $k$-bimodule for the following actions: $e_{j}\alpha e_{i}$ is equal to zero if $i\neq\mathfrak{s}(\alpha)$ or $j\neq\mathfrak{t}(\alpha)$, and is equal to $\alpha$ if $i=\mathfrak{s}(\alpha)$ and $j=\mathfrak{t}(\alpha)$.

Via the ring morphism $\mathbb{F}\rightarrow k$ that maps $1$ to $\sum_{i\in{\mathcal{Q}}_{0}}e_{i}$, the tensor $k$-algebra $T_{k}(V)$ of the $k$-bimodule $V$ is an $\mathbb{F}$-algebra isomorphic to the path algebra $\mathbb{F}\mathcal{Q}$, so that $V^{\otimes_{k}m}$ is identified with $\mathbb{F}\mathcal{Q}_{m}$, where $\mathcal{Q}_{m}$ is the set of paths of length $m$. For two arrows $\alpha$ and $\beta$, note that $\alpha\otimes_{k}\beta$ is zero if $\mathfrak{t}(\beta)\neq\mathfrak{s}(\alpha)$, and otherwise $\alpha\otimes_{k}\beta$ is identified with the composition $\alpha\beta$ of paths (where paths are written from right to left, as in [4]).

Let $R$ be a sub-$k$-bimodule of $V\otimes_{k}V\cong\mathbb{F}\mathcal{Q}_{2}$. The unital associative $k$-algebra $A=T_{k}(V)/(R)$, where $(R)$ denotes the two-sided ideal of $T_{k}(V)$ generated by $R$, is called a quadratic $k$-algebra over the finite quiver $\mathcal{Q}$. The degree induced on $A$ by the path length is called the weight, so that $A$ is a graded algebra for the weight grading. The component of weight $m$ of $A$ is denoted by $A_{m}$. Clearly, $A_{0}\cong k$ and $A_{1}\cong V$. The algebra $A$ is $\mathbb{F}$-central, meaning that the left action of $\lambda\in\mathbb{F}$ on $A$ is the same as its right action. However if there is an arrow $\alpha$ joining two distinct vertices $i$ and $j$, $A$ is not $k$-central since the left and right actions of $e_{i}$ on $\alpha$ are different. The $A$-bimodules considered in this paper are not necessarily $k$-symmetric, meaning that the left and right actions of an element of $k$ are not necessarily equal, but they are always assumed to be $\mathbb{F}$-symmetric. Setting $A^{e}=A\otimes_{\mathbb{F}}A^{op}$, any $A$-bimodule can be viewed as a left (or right) $A^{e}$-module, as usual.

For brevity, the notation $\otimes_{\mathbb{F}}$ is replaced by the unadorned tensor product $\otimes$. Similarly for the notations $\mathop{\rm Hom}\nolimits_{\mathbb{F}}$ and $\dim_{\mathbb{F}}$ abbreviated to $\mathop{\rm Hom}\nolimits$ and $\dim$. If unspecified, a vector space is an $\mathbb{F}$-vector space and a linear map is $\mathbb{F}$-linear.

The tensor product $\otimes_{k}$ is different from the unadorned tensor product $\otimes$. However, if $M$ is a right $A$-module and $N$ is a left $A$-module, then the natural linear map $Me_{i}\otimes e_{i}N\rightarrow Me_{i}\otimes_{k}e_{i}N$ is an isomorphism, so that for $a\in Me_{i}$ and $b\in e_{i}N$, we can identify $a\otimes_{k}b=a\otimes b$. Similarly, if $M$ and $N$ are $A$-bimodules, $e_{j}Me_{i}$ and $e_{i}Ne_{j}$ are $k$-bimodules, that may be viewed as left and right $k^{e}$-modules, where $k^{e}=k\otimes k$. The natural linear map $e_{j}Me_{i}\otimes e_{i}Ne_{j}\rightarrow e_{j}Me_{i}\otimes_{k^{e}}e_{i}Ne_{j}$ is an isomorphism, so for $a\in e_{j}Me_{i}$ and $b\in e_{i}Ne_{j}$, we can identify $a\otimes_{k^{e}}b=a\otimes b$. We shall freely use these identifications, without explicitly mentioning them.

Although the algebra $A$ is not $k$-central, we define its bar resolution $B(A)$ following the standard text [44] by $(A\otimes_{k}A^{\otimes_{k}\bullet}\otimes_{k}A,d)$ with

[TABLE]

for $a$, $a^{\prime}$ and $a_{1},\ldots,a_{p}$ in $A$. When $A$ is not $k$-central, the extra degeneracy is defined and is still a contracting homotopy, hence $B(A)$ is a resolution of $A$ by projective $A$-bimodules. See Lemma 2.2 below for the fact that the $A$-bimodules $A\otimes_{k}A^{\otimes_{k}p}\otimes_{k}A$ are projective.

For any $A$-bimodule $M$, Hochschild homology and cohomology are defined by

[TABLE]

[TABLE]

Given any $k$-bimodule $E$, there are well known vector space isomorphisms

[TABLE]

Taking $E=A^{\otimes_{k}p}$ and transporting $M\otimes_{A^{e}}d$ and $\mathop{\rm Hom}\nolimits_{A^{e}}(d,M)$ via these isomorphisms, we obtain the Hochschild differentials $b^{H}$ and $b_{H}$, so that

[TABLE]

[TABLE]

The Hochschild homology differential is then defined, for $m\in M$ and $a_{1},\ldots,a_{p}$ in $A$, by

[TABLE]

The Hochschild cohomology differential (including a Koszul sign in $\mathop{\rm Hom}\nolimits_{A^{e}}(d,M)$) is defined, for $f\in\mathop{\rm Hom}\nolimits_{k^{e}}(A^{\otimes_{k}p},M)$ and $a_{1},\ldots,a_{p+1}$ in $A$, by

[TABLE]

2.2 Koszul homology and cohomology

Let $A=T_{k}(V)/(R)$ be a quadratic $k$-algebra over $\mathcal{Q}$. Following [36, 40, 2, 7], the Koszul complex $K(A)$ is the subcomplex of the bar resolution $B(A)$ defined by the sub-$A$-bimodules $A\otimes_{k}W_{p}\otimes_{k}A$ of $A\otimes_{k}A^{\otimes_{k}p}\otimes_{k}A$, where $W_{0}=k$, $W_{1}=V$ and, for $p\geqslant 2$,

[TABLE]

Here $W_{p}$ is considered as a sub-$k$-bimodule of $V^{\otimes_{k}p}\subseteq A^{\otimes_{k}p}$. It is immediate that the differential $d$ of $K(A)$ is defined on $A\otimes_{k}W_{p}\otimes_{k}A$ by

[TABLE]

for $a$, $a^{\prime}$ in $A$ and $x_{1}\ldots x_{p}$ in $W_{p}$.

In this paper, we systematically follow [7] for the notation of elements of $W_{p}$. Let us recall this notation. As in (3), an arbitrary element of $W_{p}$ is denoted by a product $x_{1}\ldots x_{p}$ thought of as a sum of such products, where $x_{1},\ldots,x_{p}$ are in $V$. Moreover, regarding $W_{p}$ as a subspace of $V^{\otimes_{k}q}\otimes_{k}W_{r}\otimes_{k}V^{\otimes_{k}s}$ with $q+r+s=p$, the element $x_{1}\ldots x_{p}$ viewed in $V^{\otimes_{k}q}\otimes_{k}W_{r}\otimes_{k}V^{\otimes_{k}s}$ will be denoted by the same notation, meaning that the product $x_{q+1}\ldots x_{q+r}$ represents an element of $W_{r}$ and the other $x_{i}$ are arbitrary in $V$.

We pursue along the same lines as [7]. We present the different objects with their fundamental results more quickly. We keep the same notations as in [7] and we leave the details to the reader when they are the same as in the one vertex case.

The homology of $K(A)$ is equal to $A$ in degree [math], and to [math] in degree $1$. The quadratic algebra $A$ is said to be Koszul if the homology of $K(A)$ is [math] in any degree $>1$. Denote by $\mu:A\otimes_{k}A\rightarrow A$ the multiplication of $A$. Then $A$ is Koszul if and only if $\mu:K(A)\rightarrow A$ is a resolution of $A$. If $R=0$ and if $R=V\otimes_{k}V$, then $A$ is Koszul. Besides these extreme examples, many Koszul algebras occur in the literature, see for instance [35, 34] among many others, and it is well-known that preprojective algebras are Koszul when the graph is not Dynkin of type ADE (see Proposition 4.1 and the references in its proof).

The $A$-bimodules $A\otimes_{k}W_{p}\otimes_{k}A$ forming $K(A)$ are projective and finitely generated. Indeed, $W_{p}$ is a sub-$k$-bimodule of $V^{\otimes_{k}p}$, so that this fact is an immediate consequence of the following well known lemma (see for instance [12, Proof of Lemma 2.1]). We give an elementary proof here.

**Lemma **\theDf

Let $E$ be a $k$-bimodule.

1. \edefitit(i)

The $A$-bimodule $A\otimes_{k}E\otimes_{k}A$ is projective. 2. \edefitit(ii)

If $E$ is finite dimensional, then the $A$-bimodule $A\otimes_{k}E\otimes_{k}A$ is finitely generated.

Proof 2.1**.**

Clearly $E=\bigoplus_{i,j\in\mathcal{Q}_{0}}e_{j}Ee_{i}$. From $A=\bigoplus_{i\in\mathcal{Q}_{0}}Ae_{i}=\bigoplus_{j\in\mathcal{Q}_{0}}e_{j}A$, we deduce that the $A$-bimodule $A\otimes_{k}E\otimes_{k}A$ is isomorphic to the $A$-bimodule

[TABLE]

Considering $F_{1}$ as a sub-$A$-bimodule of $F=A\otimes E\otimes A$, we see that $F=F_{1}\oplus F_{2}$, where

[TABLE]

in which the sum is taken over the set of indices with $i_{1}\neq i_{2}$ and $i_{3}\neq i_{4}$. As the $A$-bimodule $F$ is free, we conclude that $F_{1}$ is projective. Part (ii) follows from the fact that $E$ is finite dimensional if and only if all the $(e_{j}Ee_{i})$ are finite dimensional and is left to the reader.

{Df}

For any $A$-bimodule $M$, Koszul homology and cohomology are defined by

[TABLE]

We set $\mathop{\rm HK}\nolimits_{\bullet}(A)=\mathop{\rm HK}\nolimits_{\bullet}(A,A)$ and $\mathop{\rm HK}\nolimits^{\bullet}(A)=\mathop{\rm HK}\nolimits^{\bullet}(A,A)$.

Since $K(A)$ is a complex of projective $A$-bimodules, $M\mapsto\mathop{\rm HK}\nolimits_{\bullet}(A,M)$ and $M\mapsto\mathop{\rm HK}\nolimits^{\bullet}(A,M)$ define $\delta$-functors from the category of $A$-bimodules to the category of vector spaces, that is, a short exact sequence of $A$-bimodules naturally gives rise to a long exact sequence in Koszul homology and in Koszul cohomology [44, Chapter 2]. As in [7], $\mathop{\rm HK}\nolimits_{p}(A,M)$ (respectively $\mathop{\rm HK}\nolimits^{p}(A,M)$) is isomorphic to a Hochschild hyperhomology (respectively hypercohomology) space.

The inclusion $\chi:K(A)\rightarrow B(A)$ is a morphism of complexes that induces the following morphisms of complexes

[TABLE]

[TABLE]

The linear maps $H(\tilde{\chi}):\mathop{\rm HK}\nolimits_{p}(A,M)\rightarrow\mathop{\rm HH}\nolimits_{p}(A,M)$ and $H(\chi^{\ast}):\mathop{\rm HH}\nolimits^{p}(A,M)\rightarrow\mathop{\rm HK}\nolimits^{p}(A,M)$ are always isomorphisms for $p=0$ and $p=1$, and if $A$ is Koszul they are isomorphisms for any $p$.

Taking $E=W_{p}$ in the isomorphisms (1), we get isomorphisms

[TABLE]

with differentials $b^{K}:M\otimes_{k^{e}}W_{p}\rightarrow M\otimes_{k^{e}}W_{p-1}$ and $b_{K}:\mathop{\rm Hom}\nolimits_{k^{e}}(W_{p},M)\rightarrow\mathop{\rm Hom}\nolimits_{k^{e}}(W_{p+1},M)$ given by

[TABLE]

where $m\in M$ and $x_{1}\ldots x_{p}\in W_{p}$, respectively $f\in\mathop{\rm Hom}\nolimits_{k^{e}}(W_{p},M)$ and $x_{1}\ldots x_{p+1}\in W_{p+1}$.

Note that the $k$-algebra $A$ is augmented by the natural projection $\epsilon_{A}:A\rightarrow A_{0}\cong k$. Let us examine now the particular case $M=k$, where $k$ is the $A$-bimodule defined by $\epsilon_{A}$. The action on $k$ of an element of $A_{p}$ with $p>0$ is zero, so that the Koszul differentials vanish when $M=k$. Consequently, we have the linear isomorphisms

[TABLE]

[TABLE]

In particular $\mathop{\rm HK}\nolimits^{p}(A,k)\cong\mathop{\rm Hom}\nolimits(\mathop{\rm HK}\nolimits_{p}(A,k),\mathbb{F})$, generalising [7, Proposition 2.8].

2.3 Koszul cup and cap products

Let $A=T_{k}(V)/(R)$ be a quadratic $k$-algebra over $\mathcal{Q}$. As in [7], the usual cup and cap products $\smile$ and $\frown$ in Hochschild cohomology and homology provide, by restriction from $B(A)$ to $K(A)$, the Koszul cup and cap products $\underset{K}{\smile}$ and $\underset{K}{\frown}$ in Koszul cohomology and homology. Let us give these products, expressed on Koszul cochains and chains. Let $P$, $Q$ and $M$ be $A$-bimodules. For $f\in\mathop{\rm Hom}\nolimits_{k^{e}}(W_{p},P)$, $g\in\mathop{\rm Hom}\nolimits_{k^{e}}(W_{q},Q)$ and $z=m\otimes_{k^{e}}x_{1}\ldots x_{q}\in M\otimes_{k^{e}}W_{q}$, we define $f\underset{K}{\smile}g\in\mathop{\rm Hom}\nolimits_{k^{e}}(W_{p+q},P\otimes_{A}Q)$, $f\underset{K}{\frown}z\in(P\otimes_{A}M)\otimes_{k^{e}}W_{q-p}$ and $z\underset{K}{\frown}f\in(M\otimes_{A}P)\otimes_{k^{e}}W_{q-p}$ by

[TABLE]

For any Koszul cochains $f$, $g$ $h$ and any Koszul chain $z$, we have the associativity relations

[TABLE]

inducing the same relations on Koszul classes.

As in [7, Subsection 3.1], for any Koszul cochains $f\in\mathop{\rm Hom}\nolimits_{k^{e}}(W_{p},P)$ and $g\in\mathop{\rm Hom}\nolimits_{k^{e}}(W_{q},Q)$, we have the identity

[TABLE]

so that $(\mathop{\rm Hom}\nolimits_{k^{e}}(W_{\bullet},A),b_{K},\underset{K}{\smile})$ is a DG algebra. This DG algebra will play an essential role in Section 3 and will be denoted by $\tilde{A}$. Note that $H(\tilde{A})=\mathop{\rm HK}\nolimits^{\bullet}(A)$ is a graded algebra for $\underset{K}{\smile}$. Moreover, formula (9) and analogous formulas for $b^{K}(f\stackrel{{\scriptstyle[}}{{K}}]{}{\frown}z)$ and $b^{K}(z\stackrel{{\scriptstyle[}}{{K}}]{}{\frown}f)$ show that for any $A$-bimodule $M$, $\mathop{\rm Hom}\nolimits_{k^{e}}(W_{\bullet},M)$ and $M\otimes_{k^{e}}W_{\bullet}$ are DG bimodules over $\tilde{A}$ for the actions of $\underset{K}{\smile}$ and $\underset{K}{\frown}$ respectively, so that $\mathop{\rm HK}\nolimits^{\bullet}(A,M)$ and $\mathop{\rm HK}\nolimits_{\bullet}(A,M)$ are graded $\mathop{\rm HK}\nolimits^{\bullet}(A)$-bimodules.

{Df}

Let $A=T_{k}(V)/(R)$ be a quadratic $k$-algebra over a finite quiver $\mathcal{Q}$.

1. \edefnit(i)

The general Koszul calculus of $A$ is the datum of all the spaces $\mathop{\rm HK}\nolimits^{\bullet}(A,P)$ and $\mathop{\rm HK}\nolimits_{\bullet}(A,M)$ endowed with $\underset{K}{\smile}$ and $\underset{K}{\frown}$, when the $A$-bimodules $P$ and $M$ vary. 2. \edefnit(ii)

The Koszul calculus of $A$ consists of the graded associative algebra $\mathop{\rm HK}\nolimits^{\bullet}(A)$ and of all the graded $\mathop{\rm HK}\nolimits^{\bullet}(A)$-bimodules $\mathop{\rm HK}\nolimits^{\bullet}(A,M)$ and $\mathop{\rm HK}\nolimits_{\bullet}(A,M)$, when the $A$ bimodule $M$ vary. 3. \edefnit(iii)

The restricted Koszul calculus of $A$ consists of the graded associative algebra $\mathop{\rm HK}\nolimits^{\bullet}(A)$ and of the graded $\mathop{\rm HK}\nolimits^{\bullet}(A)$-bimodule $\mathop{\rm HK}\nolimits_{\bullet}(A)$. 4. \edefnit(iv)

The scalar Koszul calculus of $A$ consists of the graded associative algebra $\mathop{\rm HK}\nolimits^{\bullet}(A,k)$ and of the graded $\mathop{\rm HK}\nolimits^{\bullet}(A,k)$-bimodule $\mathop{\rm HK}\nolimits_{\bullet}(A,k)$.

Since $\mathop{\rm HK}\nolimits^{0}(A)=Z(A)$ is the centre of the algebra $A$, the spaces $\mathop{\rm HK}\nolimits^{p}(A,M)$ and $\mathop{\rm HK}\nolimits_{p}(A,M)$ are symmetric $Z(A)$-bimodules (left and right actions coincide). However $\mathop{\rm HK}\nolimits^{p}(A,M)$ and $\mathop{\rm HK}\nolimits_{p}(A,M)$ are not $k$-bimodules in general. Indeed, $\mathop{\rm HK}\nolimits^{0}(A)=Z(A)$ itself is not a $k$-bimodule whenever there is an arrow joining two different vertices $i$ and $j$, since in this case $e_{i}1=e_{i}$ is not in $Z(A)$.

{Ex}

If $\mathcal{Q}_{1}=\emptyset$, then $V=0$ and $A$ is reduced to $k$. The (general) Koszul calculus of $k$ coincides with the (tensor) category of $k$-bimodules.

{Ex}

In order to illustrate the notation and the forthcoming results in the paper, we present the case of the preprojective algebra of type $\mathrm{A}_{3}$. This algebra is not Koszul and its Koszul calculus differs from its Hochschild calculus (see Subsection 6.6). This is a special case of the more general examples detailed in Section 6.

Let $A$ be the preprojective algebra of type $\mathrm{A}_{3}$ over $\mathbb{F}=\mathbb{C}$, that is, the $\mathbb{C}$-algebra defined by the quiver

[TABLE]

subject to the relations

[TABLE]

The algebra $A$ has dimension $10$ and a basis of $A$ over $\mathbb{C}$ is given by the elements $e_{i}$ for $0\leqslant i\leqslant 2$, $a_{i}$ and $a_{i}^{*}$ for $0\leqslant i\leqslant 1$, $a_{1}^{*}a_{1}$, $a_{1}a_{0}$ and $a_{0}^{*}a_{1}^{*}$. We then have $W_{0}=\mathbb{C}\mkern 2.0mu\overline{\mkern-2.0muQ\mkern 1.0mu}\mkern-1.0mu_{0}=\mathbb{C}\langle e_{1},e_{1},e_{2}\rangle=k$, $W_{1}=\mathbb{C}\mkern 2.0mu\overline{\mkern-2.0muQ\mkern 1.0mu}\mkern-1.0mu_{1}=\mathbb{C}\langle a_{1},a_{1},a_{0}^{*},a_{1}^{*}\rangle=V$ and $W_{2}=\mathbb{C}\langle{\sigma_{0},\sigma_{1},\sigma_{2}}\rangle=R$.

Now consider $W_{3}=(V\otimes_{k}R)\cap(R\otimes_{k}V)$, viewed inside $\mathbb{C}\mkern 2.0mu\overline{\mkern-2.0muQ\mkern 1.0mu}\mkern-1.0mu_{3}$. An element $u$ in $W_{3}$ can therefore be written as a path in $\mathbb{C}\mkern 2.0mu\overline{\mkern-2.0muQ\mkern 1.0mu}\mkern-1.0mu_{3}$ in two ways:

[TABLE]

with $\lambda_{i}$, $\lambda_{i}^{*}$, $\mu_{i}$, $\mu_{i}^{*}$ in $\mathbb{C}$. Then, in $\mathbb{C}\mkern 2.0mu\overline{\mkern-2.0muQ\mkern 1.0mu}\mkern-1.0mu_{3}$, we have

[TABLE]

so that all the coefficients $\lambda_{i}$, $\lambda_{i}^{*}$, $\mu_{i}$ and $\mu_{i}^{*}$ must be zero, hence $u=0$ and $W_{3}=0$.

Since $W_{p}=(W_{p-1}\otimes_{k}V)\cap(V\otimes_{k}W_{p-1})$ for all $p\geqslant 3$, it follows that $W_{p}=0$ for all $p\geqslant 3$. This is true for any preprojective algebra of type $\Delta$ with $\Delta$ different from $\mathrm{A}_{1}$ and $\mathrm{A}_{2}$, see Theorem 4.2.

The Koszul complex $K(A)$ is therefore

[TABLE]

with

[TABLE]

Applying $\mathop{\rm Hom}\nolimits_{A^{e}}(-,A)$ and using the natural isomorphism $\mathop{\rm Hom}\nolimits_{A^{e}}(A\otimes_{k}E\otimes_{k}A,A)\cong\mathop{\rm Hom}\nolimits_{k^{e}}(E,A)$ for any $k$-bimodule $E$ (1), we get the complex

[TABLE]

Before we describe the maps, let us note that $\mathop{\rm Hom}\nolimits_{k^{e}}(k,A)\cong\bigoplus_{i=0}^{2}e_{i}Ae_{i}$ has basis $\left\{e_{0},e_{1},e_{2},a_{1}^{*}a_{1}\right\}$, that a general element $f$ in $\mathop{\rm Hom}\nolimits_{k^{e}}(V,A)$ is defined by $f(a_{i})=\lambda_{i}a_{i}$ and $f(a_{i}^{*})=\lambda_{i}^{*}a_{i}^{*}$ for $i=0,1$ with $\lambda_{i},$ $\lambda_{i}^{*}$ in $\mathbb{C}$, and that a general element $g$ in $\mathop{\rm Hom}\nolimits_{k^{e}}(R,A)$ is defined by $g(\sigma_{i})=\alpha_{i}e_{i}$ for $i=0,2$ and $g(\sigma_{2})=\alpha_{2}e_{1}+\beta a_{1}^{*}a_{1}$ for some scalars $\alpha_{i}$ and $\beta$. Then

[TABLE]

It is then easy to see that $\mathop{\rm HK}\nolimits^{0}(A)=\mathbb{C}\langle z_{0}=1,z_{1}=a_{1}^{*}a_{1}\rangle$, that $\mathop{\rm HK}\nolimits^{1}(A)=\mathbb{C}\langle\mkern 2.0mu\overline{\mkern-2.0mu\zeta\mkern 1.0mu}\mkern-1.0mu_{0}\rangle$ with $\zeta_{0}\in\mathop{\rm Hom}\nolimits_{k^{e}}(V,A)$ defined by $\zeta_{0}(a_{i})=a_{i}$ and $\zeta_{0}(a_{i}^{*})=0$, and that $\mathop{\rm HK}\nolimits^{2}(A)=\mathbb{C}\langle\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{0},\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{1},\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{2}\rangle$ with $h_{i}\in\mathop{\rm Hom}\nolimits_{k^{e}}(R,A)$ defined by $h_{i}(\sigma_{j})=\delta_{ij}e_{i}$.

Moreover, the fundamental $1$-cocycle $\mathfrak{e}_{A}$ is equal to $2\zeta_{0}+b_{K}^{1}(2e_{0}+e_{1})$.

The Koszul cup products can easily be found using the formula (6). It follows that $\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}$ is graded commutative, that $1\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}x=x$ for any $x\in\mathop{\rm HK}\nolimits^{\bullet}(A)$ and that all other cup products are [math] in $\mathop{\rm HK}\nolimits^{\bullet}(A)$. For instance, $z_{1}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}h_{1}$ is the coboundary $b_{K}^{2}(f)$ where $f$ sends $a_{0}$ to $a_{0}$ and all other arrows to [math]. In particular, $\mathfrak{e}_{A}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}\mathfrak{e}_{A}=0$.

In order to determine the Koszul homology of $A$, we could also apply the functor $A\otimes_{A^{e}}-$ to the complex $K(A)$ and compute the homology of the complex obtained. However, we can also use our duality result, Theorem 4.3. Set $\omega_{0}=\sum_{i=0}^{2}e_{i}\otimes\sigma_{i}\in A\otimes_{k^{e}}R$. There is an isomorphism $\theta_{A}:\mathop{\rm HK}\nolimits^{\bullet}(A)\rightarrow\mathop{\rm HK}\nolimits_{2-\bullet}(A)$ given by $\mkern 2.0mu\overline{\mkern-2.0muf\mkern 1.0mu}\mkern-1.0mu\mapsto\mkern 2.0mu\overline{\mkern-2.0mu\omega\mkern 1.0mu}\mkern-1.0mu_{0}\stackrel{{\scriptstyle[}}{{K}}]{}{\frown}\mkern 2.0mu\overline{\mkern-2.0muf\mkern 1.0mu}\mkern-1.0mu$. Explicitly in our example,

1. $\theta_{A}(z_{0})=\mkern 2.0mu\overline{\mkern-2.0mu\omega\mkern 1.0mu}\mkern-1.0mu_{0}$ and $\theta_{A}(z_{1})=\mkern 2.0mu\overline{\mkern-2.0muz_{1}\otimes\sigma_{1}\mkern 1.0mu}\mkern-1.0mu$ form a basis of $\mathop{\rm HK}\nolimits_{2}(A)$;
2. $\theta_{A}(\mkern 2.0mu\overline{\mkern-2.0mu\zeta\mkern 1.0mu}\mkern-1.0mu_{0})=\mkern 2.0mu\overline{\mkern-2.0mua_{0}\otimes a_{0}^{*}+a_{1}\otimes a_{1}^{*}\mkern 1.0mu}\mkern-1.0mu$ forms a basis of $\mathop{\rm HK}\nolimits_{1}(A)$;
3. $\theta_{A}(\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{i})=\mkern 2.0mu\overline{\mkern-2.0mue_{i}\otimes e_{i}\mkern 1.0mu}\mkern-1.0mu$ for $0\leqslant i\leqslant 2$ form a basis of $\mathop{\rm HK}\nolimits_{0}(A)$.

The Koszul cap products can also be obtained using duality and they all vanish except the cap products $z_{0}\stackrel{{\scriptstyle[}}{{K}}]{}{\frown}x=x=x\stackrel{{\scriptstyle[}}{{K}}]{}{\frown}z_{0}$ for all $x\in\mathop{\rm HK}\nolimits_{\bullet}(A)$.

2.4 Fundamental formulas of Koszul calculus

Let $A=T_{k}(V)/(R)$ be a quadratic $k$-algebra over $\mathcal{Q}$. We continue to follow the one vertex case [7]. First, we define the Koszul cup and cap brackets. Let $P$, $Q$ and $M$ be $A$-bimodules, and take $f\in\mathop{\rm Hom}\nolimits_{k^{e}}(W_{p},P)$, $g\in\mathop{\rm Hom}\nolimits_{k^{e}}(W_{q},Q)$, $z\in M\otimes_{k^{e}}W_{q}$. When $P$ or $Q$ is equal to $A$, we set

[TABLE]

When $P$ or $M$ is equal to $A$, we set

[TABLE]

These brackets induce brackets on the Koszul classes.

The Koszul 1-cocycles $f:V\rightarrow M$ are called Koszul derivations with coefficients in $M$. Such an $f$ extends to a unique derivation from the $k$-algebra $A$ to the $A$-bimodule $M$, realising an isomorphism from the space of Koszul derivations with coefficients in $M$ to the space of derivations from $A$ to $M$. In particular, the Koszul 1-cocycle from $V$ to $A$ coinciding with the identity map on $V$, is sent to the Euler derivation $D_{A}$ of the graded algebra $A$. This Koszul 1-cocycle is denoted by $\mathfrak{e}_{A}$ and is called the fundamental 1-cocycle. Its Koszul class is denoted by $\overline{\mathfrak{e}}_{A}$ and is called the fundamental $1$-class. In the one vertex case, $\mathfrak{e}_{A}$ is not a coboundary if $V\neq 0$ [7], but this property does not hold in general.

**Lemma **\theDf

Let $A=T_{k}(V)/(R)$ be a quadratic $k$-algebra over $\mathcal{Q}$ with $\mathcal{Q}_{1}\neq\emptyset$. If the underlying graph of $\mathcal{Q}$ is simple, that is, it contains neither loops nor multiple edges, then $\mathfrak{e}_{A}$ is a coboundary.

Proof 2.2**.**

The 1-cocycle $\mathfrak{e}_{A}$ is a coboundary if and only if there exists a $k^{e}$-linear map $c:k\rightarrow k$ such that $\mathfrak{e}_{A}=b_{K}(c)$. Such a map is of the form $c(e_{i})=\lambda_{i}e_{i}$ with $\lambda_{i}\in\mathbb{F}$, for all $i\in\mathcal{Q}_{0}$. Then $\mathfrak{e}_{A}=b_{K}(c)$ if and only if $\lambda_{\mathfrak{t}(\alpha)}-\lambda_{\mathfrak{s}(\alpha)}=1$ for any $\alpha\in\mathcal{Q}_{1}$. The assumption on the graph means that $\mathcal{Q}$ has no loop and that given two distinct vertices, there is at most one arrow joining them. Then we can choose $\lambda_{\mathfrak{t}(\alpha)}=1$ and $\lambda_{\mathfrak{s}(\alpha)}=0$.

This proof shows that if the quiver $\mathcal{Q}$ has a loop, $\mathfrak{e}_{A}$ is not a coboundary. The same conclusion holds if $\mathop{\rm char}\nolimits\mathbb{F}\neq 2$ and $\mathcal{Q}$ contains an oriented $2$-cycle.

The following propositions are proved as [7, Theorem 3.7, Theorem 4.4, Corollary 3.10, Corollary 4.6] of the one vertex case. Formulas (12) and (13) are the fundamental formulas of Koszul calculus.

**Proposition **\theDf

Let $A=T_{k}(V)/(R)$ be a quadratic $k$-algebra over $\mathcal{Q}$. For any Koszul cochain $f$ and any Koszul chain $z$ with coefficients in an $A$-bimodule $M$, we have

[TABLE]

**Proposition **\theDf

Let $A=T_{k}(V)/(R)$ be a quadratic $k$-algebra over $\mathcal{Q}$ and let $M$ be an $A$-bimodule. For any $\alpha\in\mathop{\rm HK}\nolimits^{p}(A,M)$ with $p=0$ or $p=1$, $\beta\in\mathop{\rm HK}\nolimits^{q}(A)$ and $\gamma\in\mathop{\rm HK}\nolimits_{q}(A)$, we have the identities

[TABLE]

Identity (15) also holds if $p=q\not\in\left\{0,1\right\}$.

2.5 Higher Koszul calculus

Higher Koszul homology is the homology of the Koszul homology, and similarly for cohomology. Precisely, let $A=T_{k}(V)/(R)$ be a quadratic $k$-algebra over $\mathcal{Q}$. Formula (6) shows that the map $\mathfrak{e}_{A}\underset{K}{\smile}\mathfrak{e}_{A}:W_{2}=R\rightarrow A$ is zero. Therefore, $\mathfrak{e}_{A}\underset{K}{\smile}-$ is a cochain differential on $\mathop{\rm Hom}\nolimits_{k^{e}}(W_{\bullet},M)$, and $\overline{\mathfrak{e}}_{A}\underset{K}{\smile}-$ is a cochain differential on $\mathop{\rm HK}\nolimits^{\bullet}(A,M)$. Similarly, $\mathfrak{e}_{A}\underset{K}{\frown}-$ is a chain differential on $M\otimes_{k^{e}}W_{\bullet}$, and $\overline{\mathfrak{e}}_{A}\underset{K}{\frown}-$ is a chain differential on $\mathop{\rm HK}\nolimits_{\bullet}(A,M)$. For a $p$-cocycle $f:W_{p}\rightarrow M$ and $x_{1}\ldots x_{p+1}$ in $W_{p+1}$, we have

[TABLE]

For a $p$-cycle $z=m\otimes_{k^{e}}x_{1}\ldots x_{p}$ in $M\otimes_{k^{e}}W_{p}$, we have

[TABLE]

{Df}

Let $A=T_{k}(V)/(R)$ be a quadratic $k$-algebra over a finite quiver $\mathcal{Q}$ and let $M$ be an $A$-bimodule. The differentials $\overline{\mathfrak{e}}_{A}\underset{K}{\smile}-$ and $\overline{\mathfrak{e}}_{A}\underset{K}{\frown}-$ are denoted by $\partial_{\smile}$ and $\partial_{\frown}$. The homologies of the complexes $(\mathop{\rm HK}\nolimits^{\bullet}(A,M),\partial_{\smile})$ and $(\mathop{\rm HK}\nolimits_{\bullet}(A,M),\partial_{\frown})$ are called the higher Koszul cohomology and homology of $A$ with coefficients in $M$ and are denoted by $\mathop{\rm HK}\nolimits_{hi}^{\bullet}(A,M)$ and $\mathop{\rm HK}\nolimits^{hi}_{\bullet}(A,M)$. We set $\mathop{\rm HK}\nolimits_{hi}^{\bullet}(A)=\mathop{\rm HK}\nolimits_{hi}^{\bullet}(A,A)$ and $\mathop{\rm HK}\nolimits^{hi}_{\bullet}(A)=\mathop{\rm HK}\nolimits^{hi}_{\bullet}(A,A)$.

The higher classes of Koszul classes will be denoted between square brackets. For example, the unit $1$ of $A$ is still the unit of $\mathop{\rm HK}\nolimits^{\bullet}(A)$, and $\partial_{\smile}(1)=\overline{\mathfrak{e}}_{A}$ implies that $[\overline{\mathfrak{e}}_{A}]=0$. If $\overline{\mathfrak{e}}_{A}\neq 0$, the unit of $\mathop{\rm HK}\nolimits^{\bullet}(A)$ does not survive in higher Koszul cohomology.

As in the one vertex case, the actions of the Koszul cup and cap products of $\mathop{\rm HK}\nolimits^{\bullet}(A)$ on $\mathop{\rm HK}\nolimits^{\bullet}(A,M)$ and $\mathop{\rm HK}\nolimits_{\bullet}(A,M)$ induce actions on higher cohomology and homology. Thus $\mathop{\rm HK}\nolimits_{hi}^{\bullet}(A)$ is a graded algebra, and $\mathop{\rm HK}\nolimits_{hi}^{\bullet}(A,M)$, $\mathop{\rm HK}\nolimits^{hi}_{\bullet}(A,M)$ are graded $\mathop{\rm HK}\nolimits_{hi}^{\bullet}(A)$-bimodules, constituting the higher Koszul calculus of $A$. If $\overline{\mathfrak{e}}_{A}=0$, the higher Koszul calculus coincides with the Koszul calculus. It is the case when $A=k$ as in Example 2.3.

For $M=k$, $\mathfrak{e}_{A}\underset{K}{\smile}-$ and $\mathfrak{e}_{A}\underset{K}{\frown}-$ vanish, so that the higher scalar Koszul calculus coincides with the scalar Koszul calculus. Proposition 3.12 in [7] generalises immediately as follows.

**Proposition **\theDf

Let $A=T_{k}(V)/(R)$ be a quadratic $k$-algebra over $\mathcal{Q}$ and let $M$ be an $A$-bimodule. Then $\mathop{\rm HK}\nolimits_{hi}^{0}(A,M)$ is the space of elements $u$ in $Z(M)$ such that there exists $v\in M$ satisfying $u.a=v.a-a.v$ for any $a$ in $\mathcal{Q}_{1}$.

2.6 Grading the restricted Koszul calculus by the weight

A Koszul $p$-cochain $f:W_{p}\rightarrow A_{m}$ is said to be homogeneous of weight $m$. Since $\mathcal{Q}_{1}$ is finite, the spaces $W_{p}$ are finite dimensional, thus the space of Koszul cochains $\mathop{\rm Hom}\nolimits_{k^{e}}(W_{\bullet},A)$ is $\mathbb{N}\times\mathbb{N}$-graded by the biweight $(p,m)$, where $p$ is called the homological weight and $m$ is called the coefficient weight. If $f:W_{p}\rightarrow A_{m}$ and $g:W_{q}\rightarrow A_{n}$ are homogeneous of biweights $(p,m)$ and $(q,n)$ respectively, then $f\underset{K}{\smile}g:W_{p+q}\rightarrow A_{m+n}$ is homogeneous of biweight $(p+q,m+n)$. Moreover $b_{K}$ is homogeneous of biweight $(1,1)$ and the algebra $\mathop{\rm HK}\nolimits^{\bullet}(A)$ is $\mathbb{N}\times\mathbb{N}$-graded by the biweight. The homogeneous component of biweight $(p,m)$ of $\mathop{\rm HK}\nolimits^{\bullet}(A)$ is denoted by $\mathop{\rm HK}\nolimits^{p}(A)_{m}$. Since

[TABLE]

the algebra $\mathop{\rm HK}\nolimits_{hi}^{\bullet}(A)$ is $\mathbb{N}\times\mathbb{N}$-graded by the biweight, and its $(p,m)$-component is denoted by $\mathop{\rm HK}\nolimits_{hi}^{p}(A)_{m}$. From Proposition 2.5, we deduce the following.

**Proposition **\theDf

Let $A=T_{k}(V)/(R)$ be a quadratic $k$-algebra over $\mathcal{Q}$. Assume that $A$ is finite dimensional. Let $\max$ be the highest $m$ such that $A_{m}\neq 0$. Then $\mathop{\rm HK}\nolimits_{hi}^{0}(A)_{\max}$ is isomorphic to the space spanned by the cycles of $\mathcal{Q}$ of length $\max$.

Similarly, a Koszul $q$-chain $z$ in $A_{n}\otimes_{k^{e}}W_{q}$ is said to be homogeneous of weight $n$. The space of Koszul chains $A\otimes_{k^{e}}W_{\bullet}$ is $\mathbb{N}\times\mathbb{N}$-graded by the biweight $(q,n)$, where $q$ is called the homological weight and $n$ is called the coefficient weight. Moreover $b^{K}$ is homogeneous of biweight $(-1,1)$ and the space $\mathop{\rm HK}\nolimits_{\bullet}(A)$ is $\mathbb{N}\times\mathbb{N}$-graded by the biweight. The homogeneous component of biweight $(q,n)$ of $\mathop{\rm HK}\nolimits_{\bullet}(A)$ is denoted by $\mathop{\rm HK}\nolimits_{q}(A)_{n}$. Since

[TABLE]

the space $\mathop{\rm HK}\nolimits^{hi}_{\bullet}(A)$ is $\mathbb{N}\times\mathbb{N}$-graded by the biweight, and its $(q,n)$-component is denoted by $\mathop{\rm HK}\nolimits^{hi}_{q}(A)_{n}$.

If $f:W_{p}\rightarrow A_{m}$ and $z\in A_{n}\otimes_{k^{e}}W_{q}$ are homogeneous of biweights $(p,m)$ and $(q,n)$ respectively, then $f\underset{K}{\frown}z$ and $z\underset{K}{\frown}f$ are homogeneous of biweight $(q-p,m+n)$ where

[TABLE]

and $z=a\otimes_{k^{e}}x_{1}\ldots x_{q}$. The $\mathop{\rm Hom}\nolimits_{k^{e}}(W_{\bullet},A)$-bimodule $A\otimes_{k^{e}}W_{\bullet}$, the $\mathop{\rm HK}\nolimits^{\bullet}(A)$-bimodule $\mathop{\rm HK}\nolimits_{\bullet}(A)$ and the $\mathop{\rm HK}\nolimits_{hi}^{\bullet}(A)$-bimodule $\mathop{\rm HK}\nolimits^{hi}_{\bullet}(A)$ are thus $\mathbb{N}\times\mathbb{N}$-graded by the biweight. The proof of the following is left to the reader.

**Proposition **\theDf

Let $A=T_{k}(V)/(R)$ be a quadratic $k$-algebra over $\mathcal{Q}$. We have

[TABLE]

Moreover $\mathop{\rm HK}\nolimits_{0}(A)_{1}\cong\mathop{\rm HK}\nolimits_{1}(A)_{0}$ is isomorphic to the space spanned by the loops of $\mathcal{Q}$, and $\partial_{\frown}:\mathop{\rm HK}\nolimits_{1}(A)_{0}\rightarrow\mathop{\rm HK}\nolimits_{0}(A)_{1}$ identifies with the identity map on this space. As a consequence,

[TABLE]

2.7 Invariance of Koszul calculus

In [6], the first author proved that the Koszul calculus of an $N$-homogeneous algebra $A$ over a field $k$ only depends on the structure of associative algebra of $A$, independently of any presentation $A=T_{k}(V)/(R)$ of $A$ as an $N$-homogeneous algebra. This result was based on an isomorphism lemma due to Bell and Zhang [3]. In the quadratic case $N=2$, we are going to extend this Koszul calculus invariance to any quadratic quiver algebra. For that, we shall use an extension of the isomorphism lemma to quiver algebras with homogeneous relations, due to Gaddis [25].

Let $\mathcal{Q}$ and $\mathcal{Q^{\prime}}$ be finite quivers, and $\mathbb{F}$ be a field. We introduce the commutative rings $k=\mathbb{F}\mathcal{Q}_{0}$ and $k^{\prime}=\mathbb{F}\mathcal{Q^{\prime}}_{0}$, the $k$-bimodule $V=\mathbb{F}\mathcal{Q}_{1}$ and the $k^{\prime}$-bimodule $V^{\prime}=\mathbb{F}\mathcal{Q}^{\prime}_{1}$. As explained in Subsection 2.1, we make the identifications of graded algebras $T_{k}(V)\cong\mathbb{F}\mathcal{Q}$ and $T_{k^{\prime}}(V^{\prime})\cong\mathbb{F}\mathcal{Q^{\prime}}$. We are interested in the graded $\mathbb{F}$-algebra isomorphisms $u:T_{k}(V)\rightarrow T_{k^{\prime}}(V^{\prime})$ given by a ring isomorphim $u_{0}:k\rightarrow k^{\prime}$ and by a $k$-bimodule isomorphism $u_{1}:V\rightarrow V^{\prime}$, where $V^{\prime}$ is a $k$-bimodule via $u_{0}$. By [25, Lemma 4], this implies that $u_{0}$ maps $\mathcal{Q}_{0}$ to $\mathcal{Q^{\prime}}_{0}$, and the bijection $\mathcal{Q}_{0}\rightarrow\mathcal{Q^{\prime}}_{0}$ induced by $u_{0}$ transforms the adjacency matrix of $\mathcal{Q}$ into the adjacency matrix of $\mathcal{Q^{\prime}}$.

Let us fix a sub-$k$-bimodule $R$ of $V\otimes_{k}V$ and a sub-$k^{\prime}$-bimodule $R^{\prime}$ of $V^{\prime}\otimes_{k^{\prime}}V^{\prime}$. We define the graded $k$-algebra $A=T_{k}(V)/(R)$ and the graded $k^{\prime}$-algebra $A^{\prime}=T_{k^{\prime}}(V^{\prime})/(R^{\prime})$. Following the terminology of the one vertex case, a graded $\mathbb{F}$-algebra isomorphism $u:A\rightarrow A^{\prime}$ is called a Manin isomorphism if $u$ is defined by a ring isomorphism $u_{0}:k\rightarrow k^{\prime}$ (so $V^{\prime}$ is a $k$-bimodule via $u_{0}$), and by a $k$-bimodule isomorphism $u_{1}:V\rightarrow V^{\prime}$, such that the $k$-bimodule isomorphism $u_{1}^{\otimes_{k}2}:V^{\otimes_{k}2}\rightarrow V^{\prime\otimes_{k^{\prime}}2}$ satisfies $u_{1}^{\otimes_{k}2}(R)=R^{\prime}$. In particular, $u$ is an isomorphism of the augmented $k$-algebra $A$ to the augmented $k^{\prime}$-algebra $A^{\prime}$, the augmentations being the projections $A\rightarrow A_{0}\cong k$ and $A^{\prime}\rightarrow A^{\prime}_{0}\cong k^{\prime}$.

As in [6], for any $A$-bimodule $M$, the Manin isomorphism $u$ naturally defines an isomorphism of complexes from $(M\otimes_{k^{e}}W_{\bullet},b^{K})$ to $(M\otimes_{k^{\prime e}}W^{\prime}_{\bullet},b^{K})$, where $M$ is an $A^{\prime}$-bimodule via $u$, inducing natural isomorphisms $\mathop{\rm HK}\nolimits_{\bullet}(A,M)\cong\mathop{\rm HK}\nolimits_{\bullet}(A^{\prime},M)$. Similarly, $u$ induces natural isomorphisms $\mathop{\rm HK}\nolimits^{\bullet}(A^{\prime},M)\cong\mathop{\rm HK}\nolimits^{\bullet}(A,M)$. It is clear from the definitions in Subsection 2.3 that these isomorphisms respect the Koszul cup and cap products. To summarise all these properties, we say that a Manin isomorphism induces isomorphic (general) Koszul calculi. Since $u_{1}(\mathfrak{e}_{A})=\mathfrak{e}_{A^{\prime}}$ by functoriality, it also induces isomorphic higher Koszul calculi.

Using Gaddis’s theorem [25, Theorem 5], we can now prove ungraded invariance. Let $C$ be a commutative ring. Let $A$ be an augmented associative $C$-algebra (not necessarily $C$-central) having a quadratic quiver algebra presentation $B$, meaning that the augmented $C$-algebra $A$ is isomorphic to a quadratic $k$-algebra $B=T_{k}(V)/(R)$ over a finite quiver $\mathcal{Q}$, naturally augmented over $k$ by the projection $B\rightarrow B_{0}\cong k$. This implies that the ring $C$ is isomorphic to $k\cong\mathbb{F}\mathcal{Q}_{0}$. Then we can define the (general) Koszul calculus of $A$ as being the (general) Koszul calculus of $B$. Indeed, if $B^{\prime}=T_{k^{\prime}}(V^{\prime})/(R^{\prime})$ over a finite quiver $\mathcal{Q^{\prime}}$ is another quadratic quiver algebra presentation of $A$, the ungraded augmented $k$-algebra $B$ is isomorphic to the ungraded augmented $k^{\prime}$-algebra $B^{\prime}$. By Gaddis’s theorem, there exists a Manin isomorphism from $B$ to $B^{\prime}$, thus the (general) Koszul calculi of $B$ and $B^{\prime}$ are isomorphic by Manin invariance. The higher Koszul calculus of $A$ is also defined as being the higher Koszul calculus of $B$.

2.8 Comparing Koszul (co)homology with Hochschild (co)homology in degree 2

Let $A=T_{k}(V)/(R)$ be a quadratic $k$-algebra over $\mathcal{Q}$. Recall that, for $p=0$ and $p=1$, we have linear isomorphisms $\mathop{\rm HK}\nolimits_{p}(A,M)\cong\mathop{\rm HH}\nolimits_{p}(A,M)$ and $\mathop{\rm HH}\nolimits^{p}(A,M)\cong\mathop{\rm HK}\nolimits^{p}(A,M)$ (Subsection 2.2). It is no longer true if $p\geqslant 2$ and if $A$ is an arbitrary non-Koszul algebra. Preprojective algebras of Dynkin type will give infinitely many counterexamples when $p=2$. However, in general, we can compare the Koszul and Hochschild spaces when $p=2$, by providing a surjection $\mathop{\rm HK}\nolimits_{2}(A,M)\rightarrow\mathop{\rm HH}\nolimits_{2}(A,M)$ and an injection $\mathop{\rm HH}\nolimits^{2}(A,M)\rightarrow\mathop{\rm HK}\nolimits^{2}(A,M)$. To prove that, we use a minimal projective resolution of the graded $k$-algebra $A$, described as follows.

As in the one vertex case [7], we know that, in the category of graded $A$-bimodules, $A$ has a minimal projective resolution $P(A)$ whose component of homological degree $p$ can be written as $A\otimes_{k}E_{p}\otimes_{k}A$, where $E_{p}$ is a weight-graded $k$-bimodule. Then $P_{l}(A)=P(A)\otimes_{A}k$ (respectively $P_{r}(A)=k\otimes_{A}P(A)$) is a minimal projective resolution of the graded left (respectively right) $A$-module $k$ (see for instance [5]) and the differential $\delta$ of $P(A)$ is the graded sum of the differentials $\delta\otimes_{A}\mathop{\rm id}\nolimits_{k}$ and $\mathop{\rm id}\nolimits_{k}\otimes_{A}\delta$ naturally extended to $P(A)$.

Define the left (respectively right) Koszul complex $K_{l}(A)=K(A)\otimes_{A}k$ (respectively $K_{r}(A)=k\otimes_{A}K(A)$). Now using [2, Subsections 2.7 and 2.8], [35, Chapter 1, Proposition 3.1] adapted to the case where $k$ is a semisimple ring (rather than a field), and the construction of $\mathop{\rm Ext}\nolimits_{A}(k,k)$ from the resolutions $P_{l}(A)$ and $P_{r}(A)$, we can show that $K_{l}(A)$ (respectively $K_{r}(A)$) is isomorphic as a left (respectively right) $A$-module to the diagonal part of the graded resolution $P_{l}(A)$ (respectively $P_{r}(A)$). We know (see for instance [40, Section 3]) that the differential $d$ of $K(A)$ is the graded sum of the differentials $d\otimes_{A}\mathop{\rm id}\nolimits_{k}$ and $\mathop{\rm id}\nolimits_{k}\otimes_{A}d$ naturally extended to $K(A)$. Thus the inclusions $A\otimes_{k}W_{p}\otimes_{k}A\hookrightarrow A\otimes_{k}E_{p}\otimes_{k}A$ constitute an inclusion map $\iota:K(A)\hookrightarrow P(A)$ of weight-graded $A$-bimodule complexes. So we can view the complex $K(A)$ as the diagonal part of the weight-graded resolution $P(A)$, and $A$ is Koszul if and only if $P(A)=K(A)$. The beginning of $P(A)$ coincides with $K(A)$, that is, $E_{0}=k$, $E_{1}=V$, $E_{2}=R$, and the differential $\delta$ of $P(A)$ coincides with the differential $d$ of $K(A)$ in degrees 1 and 2.

For any $A$-bimodule $M$, $\iota$ induces $\tilde{\iota}=M\otimes_{A^{e}}\iota$ and $\iota^{\ast}=\mathop{\rm Hom}\nolimits_{A^{e}}(\iota,M)$ decomposed in $\tilde{\iota}_{p}:M\otimes_{k^{e}}W_{p}\rightarrow M\otimes_{k^{e}}E_{p}$ and $\iota^{\ast}_{p}:\mathop{\rm Hom}\nolimits_{k^{e}}(E_{p},M)\rightarrow\mathop{\rm Hom}\nolimits_{k^{e}}(W_{p},M)$. The linear maps

[TABLE]

are isomorphisms for $p=0$ and $p=1$, and for any $p$ if $A$ is Koszul. Since $\iota_{p}$ is an identity map for $p=0,1,2$, $\tilde{\iota}_{p}$ and $\iota^{\ast}_{p}$ are also identity maps for the same $p$’s. Therefore, setting $\tilde{\delta}=M\otimes_{A^{e}}\delta$ and $\delta^{\ast}=\mathop{\rm Hom}\nolimits_{A^{e}}(\delta,M)$, we have the commutative diagrams

[TABLE]

[TABLE]

where $\tilde{\iota}_{3}$ is injective and $\iota^{\ast}_{3}$ is surjective (the ring $k^{e}\cong\mathbb{F}^{|\mathcal{Q}_{0}|^{2}}$ is semisimple), so that we obtain the following.

**Proposition **\theDf

Let $A=T_{k}(V)/(R)$ be a quadratic $k$-algebra over $\mathcal{Q}$. For any $M$,

1. \edefitit(i)

$H(\tilde{\iota})_{2}:\mathop{\rm HK}\nolimits_{2}(A,M)\rightarrow\mathop{\rm HH}\nolimits_{2}(A,M)$* is surjective with kernel isomorphic to $\mathop{\rm Im}\nolimits(\tilde{\delta}_{3})/\mathop{\rm Im}\nolimits(b^{K}_{3})$,* 2. \edefitit(ii)

$H(\iota^{\ast})_{2}:\mathop{\rm HH}\nolimits^{2}(A,M)\rightarrow\mathop{\rm HK}\nolimits^{2}(A,M)$* is injective with image isomorphic to $\mathop{\rm Ker}\nolimits(\delta^{\ast}_{3})/\mathop{\rm Im}\nolimits(b_{K}^{2})$.*

We can be more specific when $M=A$, by using the weight grading (Subsection 2.6). Unlike the Koszul differentials $b^{K}$ and $b_{K}$, the Hochschild differentials $b^{H}$ and $b_{H}$ are not homogeneous for the coefficient weight, but only for the total weight. The grading of $\mathop{\rm HH}\nolimits_{p}(A)$ and $\mathop{\rm HH}\nolimits^{p}(A)$ for the total weight $t$ is denoted by $\mathop{\rm HH}\nolimits_{p}(A)_{t}$ and $\mathop{\rm HH}\nolimits^{p}(A)_{t}$. Denote the weight of a homogeneous element $a$ of $A$ by $|a|$. Recall that the total weight of a homogeneous $p$-chain $z=a\otimes_{k^{e}}(a_{1}\ldots a_{p})$ is equal to $t=|a|+|a_{1}|+\ldots+|a_{p}|$, and the total weight of a homogeneous $p$-cochain $f$ mapping $a_{1}\ldots a_{p}$ to an element of $A_{m}$ is equal to $t=m-|a_{1}|-\ldots-|a_{p}|$. Then $H(\tilde{\iota})_{2}$ is homogeneous from the coefficient weight $r$ to the total weight $r+2$, while $H(\iota^{\ast})_{2}$ is homogeneous from the total weight $r-2$ to the coefficient weight $r$.

**Corollary **\theDf

Let $A=T_{k}(V)/(R)$ be a quadratic $k$-algebra over $\mathcal{Q}$.

1. \edefitit(i)

$H(\tilde{\iota})_{2}$* is an isomorphism from $\mathop{\rm HK}\nolimits_{2}(A)_{r}$ to $\mathop{\rm HH}\nolimits_{2}(A)_{r+2}$ if $r=0$ and $r=1$.* 2. \edefitit(ii)

Assume that $A$ is finite dimensional. Let $\max$ be the highest $m$ such that $A_{m}\neq 0$. Then $H(\iota^{\ast})_{2}$ is an isomorphism from $\mathop{\rm HH}\nolimits^{2}(A)_{r-2}$ to $\mathop{\rm HK}\nolimits^{2}(A)_{r}$ if $r=\max$ and $r=\max-1$.

Proof 2.3**.**

Denote by $E_{p,m}$ the homogeneous component of weight $m$ of $E_{p}$. Since $E_{3,2}=0$ and $E_{3,3}=W_{3}$, both maps $\tilde{\delta}_{3}$ and $b^{K}_{3}$ vanish on the component of total weight 2 of $A\otimes_{k^{e}}E_{3}$, while on that of total weight 3, they coincide with the inclusion map of $W_{3}$ into $V\otimes_{k^{e}}R$. Then we deduce (i) from (i) of the proposition.

Under the assumptions of (ii), if $f:R\rightarrow A_{\max}$, then $b_{K}^{3}(f)=0$. Moreover, any other component of $\delta^{\ast}_{3}(f)$ mapping $E_{3,m}$ to $A_{\max+m-2}=0$ vanishes as well. Thus $\delta^{\ast}_{3}(f)=0$, and we conclude by (ii) of the proposition. The same proof works if $f:R\rightarrow A_{\max-1}$ since $\delta^{\ast}_{3}(f)$ is then reduced to a map $W_{3}\rightarrow A_{\max}$ coinciding with $b_{K}^{3}(f)$.

3 A right action on the Koszul calculus

This section presents an important tool which we use in Section 5 to adapt the known definition of Calabi-Yau algebras due to Ginzburg to the context of quadratic quiver algebras endowed with the Koszul calculus. The idea is to put together two compatible bimodule actions on Koszul chains and cochains : the action of the quadratic quiver algebra $A$ and the action of the associated DG algebra $\tilde{A}$ defined just before Definition 2.3.

3.1 Compatibility

**Lemma **\theDf

Let $A$ and $B$ be unital associative $\mathbb{F}$-algebras. Let $M$ be an $A$-bimodule (hence the induced $\mathbb{F}$-bimodule is symmetric). Assume that $M$ is a right $B$-module such that the actions of $\mathbb{F}$ induced on $M$ by $A$ and by $B$ are the same. Let $A^{e}=A\otimes_{\mathbb{F}}A^{op}$ be the enveloping algebra. The following are equivalent.

1. \edefitit(i)

Viewing $M$ as a left $A^{e}$-module, $M$ is an $A^{e}$-$B$-bimodule. 2. \edefitit(ii)

Viewing $M$ as a right $A^{e}$-module, the right actions of $A^{e}$ and $B$ on $M$ commute. 3. \edefitit(iii)

$M$* is an $A$-$B$-bimodule and the right actions of $A$ and $B$ on $M$ commute.*

The proof is straightforward. Under the assumptions of the lemma and if the equivalent assertions hold, we say that the right action of $B$ on $M$ is compatible with the $A$-bimodule $M$.

{Ex}

With $B=M=A^{e}$, $A^{e}$ is a natural $A^{e}$-$A^{e}$-bimodule for the multiplication of the $\mathbb{F}$-algebra $A^{e}$. Recall that the left $A^{e}$-module $A^{e}$ is isomorphic to the $A$-bimodule $A\stackrel{{\scriptstyle o}}{{\otimes}}A$ for the outer action $(a\otimes b).(\alpha\otimes\beta)=(a\alpha)\otimes(\beta b)$, while the right $A^{e}$-module $A^{e}$ is isomorphic to the $A$-bimodule $A\stackrel{{\scriptstyle i}}{{\otimes}}A$ for the inner action $(\alpha\otimes\beta).(a\otimes b)=(\alpha a)\otimes(b\beta)$.

3.2 DG bimodules over the DG algebra $\tilde{A}$

Let $A=T_{k}(V)/(R)$ be a quadratic $k$-algebra over $\mathcal{Q}$. Fix a unital associative $\mathbb{F}$-algebra $B$ and an $A$-bimodule $M$. We assume that $M$ is a right $B$-module compatible with the $A$-bimodule structure. Then the space $M\otimes_{k^{e}}W_{q}$ is a right $B$-module for the action of $b\in B$ on $z=m\otimes_{k^{e}}x_{1}\ldots x_{q}\in M\otimes_{k^{e}}W_{q}$ defined by

[TABLE]

It is well-defined since $(\lambda m\mu).b=\lambda(m.b)\mu$ for any $\lambda$ and $\mu$ in $k$. From (4) and (7), we check that $b^{K}$ and $\mathfrak{e}_{A}\underset{K}{\frown}-$ are $B$-linear. Thus $\mathop{\rm HK}\nolimits_{\bullet}(A,M)$ and $\mathop{\rm HK}\nolimits^{hi}_{\bullet}(A,M)$ are graded right $B$-modules.

Just before Definition 2.3, we have associated to $A$ the $\mathbb{F}$-central DG algebra

[TABLE]

whose grading is given by the cohomological degree of cochains, whose differential is $b_{K}$ and whose multiplication is $\underset{K}{\smile}$. We have also mentioned that $\mathop{\rm Hom}\nolimits_{k^{e}}(W_{\bullet},M)$ and $M\otimes_{k^{e}}W_{\bullet}$ are DG bimodules over $\tilde{A}$ for the actions of $\underset{K}{\smile}$ and $\underset{K}{\frown}$ respectively, so that $\mathop{\rm HK}\nolimits^{\bullet}(A,M)$ and $\mathop{\rm HK}\nolimits_{\bullet}(A,M)$ are graded $\mathop{\rm HK}\nolimits^{\bullet}(A)$-bimodules.

For any $k$-bimodule morphism $f:W_{p}\rightarrow A$, we verify that

[TABLE]

so that the right action of $B$ on $M\otimes_{k^{e}}W_{\bullet}$ is compatible with the $\tilde{A}$-bimodule structure. Therefore the right action of $B$ on $\mathop{\rm HK}\nolimits_{\bullet}(A,M)$ and on $\mathop{\rm HK}\nolimits^{hi}_{\bullet}(A,M)$ is compatible with the structure of $\mathop{\rm HK}\nolimits^{\bullet}(A)$-bimodule and of $\mathop{\rm HK}\nolimits_{hi}^{\bullet}(A)$-bimodule respectively. Let us sum up what we have obtained at the level of complexes.

**Proposition **\theDf

Let $A=T_{k}(V)/(R)$ be a quadratic $k$-algebra over $\mathcal{Q}$, let $B$ be a unital associative $\mathbb{F}$-algebra and let $M$ be an $A$-bimodule. We assume that $M$ is a right $B$-module and that this structure is compatible with the $A$-bimodule structure. Denote by $\text{{Mod}-}B$ the category of right $B$-modules. Then

1. \edefitit(i)

the complex $(M\otimes_{k^{e}}W_{\bullet},b^{K})$ is a complex in $\text{{Mod}-}B$, 2. \edefitit(ii)

the complex $(M\otimes_{k^{e}}W_{\bullet},b^{K})$ is a DG-bimodule over the $\mathbb{F}$-central DG-algebra $\tilde{A}$, 3. \edefitit(iii)

the right action of $B$ and the bimodule action of $\tilde{A}$ on $M\otimes_{k^{e}}W_{\bullet}$ are compatible.

In this situation, following Yekutieli [45, Definition 3.8.1], we say that $M\otimes_{k^{e}}W_{\bullet}$ is a DG $\tilde{A}$-bimodule in the abelian category $\text{{Mod}-}B$.

In order to reflect the fact that the right action of $B$ and the bimodule actions of $\mathop{\rm HK}\nolimits^{\bullet}(A)$ and $\mathop{\rm HK}\nolimits_{hi}^{\bullet}(A)$ on $\mathop{\rm HK}\nolimits_{\bullet}(A,M)$ and $\mathop{\rm HK}\nolimits^{hi}_{\bullet}(A,M)$ respectively are compatible, we shall also say that $\mathop{\rm HK}\nolimits_{\bullet}(A,M)$ is a graded $\mathop{\rm HK}\nolimits^{\bullet}(A)$-bimodule in $\text{{Mod}-}B$, and that $\mathop{\rm HK}\nolimits^{hi}_{\bullet}(A,M)$ is a graded $\mathop{\rm HK}\nolimits_{hi}^{\bullet}(A)$-bimodule in $\text{{Mod}-}B$. By Lemma 3.1, it is equivalent to saying that $\mathop{\rm HK}\nolimits_{\bullet}(A,M)$ is a graded $\mathop{\rm HK}\nolimits^{\bullet}(A)^{e}$-$B$-bimodule, similarly for $\mathop{\rm HK}\nolimits^{hi}_{\bullet}(A,M)$.

Similarly, $\mathop{\rm Hom}\nolimits_{k^{e}}(W_{\bullet},M)$ is a right $B$-module for the action of $b$ on $f:W_{p}\rightarrow M$ defined by

[TABLE]

Then $b_{K}$ and $\mathfrak{e}_{A}\underset{K}{\smile}-$ are $B$-linear, so that $\mathop{\rm HK}\nolimits^{\bullet}(A,M)$ and $\mathop{\rm HK}\nolimits_{hi}^{\bullet}(A,M)$ are graded right $B$-modules. For $g:W_{q}\rightarrow A$, we have

[TABLE]

We obtain an analogue of Proposition 3.2, that is, $\mathop{\rm Hom}\nolimits_{k^{e}}(W_{\bullet},M)$ is a DG $\tilde{A}$-bimodule in $\text{{Mod}-}B$.

**Proposition **\theDf

We keep the notation and assumptions of the previous proposition. Then

1. \edefitit(i)

the complex $\mathop{\rm Hom}\nolimits_{k^{e}}(W_{\bullet},M)$ is a complex in $\text{{Mod}-}B$, 2. \edefitit(ii)

the complex $\mathop{\rm Hom}\nolimits_{k^{e}}(W_{\bullet},M)$ is a DG $\tilde{A}$-bimodule, 3. \edefitit(iii)

the right action of $B$ and the bimodule action of $\tilde{A}$ are compatible on $\mathop{\rm Hom}\nolimits_{k^{e}}(W_{\bullet},M)$.

Therefore $\mathop{\rm HK}\nolimits^{\bullet}(A,M)$ is a graded $\mathop{\rm HK}\nolimits^{\bullet}(A)$-bimodule in $\text{{Mod}-}B$, and $\mathop{\rm HK}\nolimits_{hi}^{\bullet}(A,M)$ is a graded $\mathop{\rm HK}\nolimits_{hi}^{\bullet}(A)$-bimodule in $\text{{Mod}-}B$.

3.3 Application to the Koszul complex $K(A)$

Let us specialise to $B=M=A^{e}$ as in Example 3.1. Then $M=A\stackrel{{\scriptstyle o}}{{\otimes}}A$ is a left $A^{e}$-module for the outer structure, and a right $A^{e}$-module for the inner structure. Our aim is to identify the $A$-bimodule complex $K(A)$ with the complex $((A\stackrel{{\scriptstyle o}}{{\otimes}}A)\otimes_{k^{e}}W_{\bullet},b^{K})$ endowed with the right action of $A^{e}$. The statement is the following.

**Proposition **\theDf

Let $A=T_{k}(V)/(R)$ be a quadratic $k$-algebra over $\mathcal{Q}$.

1. \edefitit(i)

For any $q\geqslant 0$, the bilinear map $\varphi_{q}:(A\stackrel{{\scriptstyle o}}{{\otimes}}A)\times W_{q}\rightarrow A\otimes_{k}W_{q}\otimes_{k}A$ defined by

[TABLE]

induces an isomorphism $\tilde{\varphi}_{q}:(A\stackrel{{\scriptstyle o}}{{\otimes}}A)\otimes_{k^{e}}W_{q}\rightarrow A\otimes_{k}W_{q}\otimes_{k}A$. 2. \edefitit(ii)

The direct sum $\tilde{\varphi}$ of the maps $\tilde{\varphi}_{q}$ is an isomorphism from the complex $((A\stackrel{{\scriptstyle o}}{{\otimes}}A)\otimes_{k^{e}}W_{\bullet},b^{K})$ to the Koszul complex $(K(A),d)$. 3. \edefitit(iii)

The isomorphism $\tilde{\varphi}$ is right $A^{e}$-linear.

Proof 3.1**.**

The $A$-bimodule $A\stackrel{{\scriptstyle o}}{{\otimes}}A$ is a $k$-bimodule for the actions $\lambda(\alpha\otimes\beta)\mu=\lambda\alpha\otimes\beta\mu$, with $\alpha$ and $\beta$ in $A$, $\lambda$ and $\mu$ in $k$, thus it is a right $k^{e}$-module for $(\alpha\otimes\beta)(\lambda\otimes\mu)=\mu\alpha\otimes\beta\lambda$. Then it is easy to check that

[TABLE]

proving the existence of $\tilde{\varphi}_{q}$. We define similarly an inverse linear map, therefore $\tilde{\varphi}_{q}$ is an isomorphism, which gives (i).

Let us show that $\tilde{\varphi}$ is a morphism of complexes. From

[TABLE]

we get

[TABLE]

whose right-hand side is equal to $d(\beta\otimes_{k}x_{1}\ldots x_{p}\otimes_{k}\alpha)$, as expected.

Let us prove (iii). Here the $A$-bimodule $A\otimes_{k}W_{q}\otimes_{k}A$ is seen as a right $A^{e}$-module. For $z=(\alpha\otimes\beta)\otimes_{k^{e}}x_{1}\ldots x_{q}$ and $a$, $b$ in $A$, we have

[TABLE]

*therefore $\tilde{\varphi}_{q}$ is $A^{e}$-linear. *

So $\tilde{\varphi}$ is an isomorphism from the $A$-bimodule complex $((A\stackrel{{\scriptstyle o}}{{\otimes}}A)\otimes_{k^{e}}W_{\bullet},b^{K})$ whose $A$-bimodule structure is the inner one, to the $A$-bimodule complex $K(A)$. Denote by $A\text{-{Bimod}}$ the category of $A$-bimodules. According to Proposition 3.2, $(A\stackrel{{\scriptstyle o}}{{\otimes}}A)\otimes_{k^{e}}W_{\bullet}$ is a DG $\tilde{A}$-bimodule in $A\text{-{Bimod}}$. We transport this structure via $\tilde{\varphi}$ and we obtain.

**Proposition **\theDf

Let $A=T_{k}(V)/(R)$ be a quadratic $k$-algebra over $\mathcal{Q}$. Then the Koszul complex $K(A)$ is a DG $\tilde{A}$-bimodule in $A\text{-{Bimod}}$.

This DG bimodule will play an essential role in the generalisations of Calabi-Yau algebras (Sections 4 and 5). Moreover, $H(K(A))$ is a graded $\mathop{\rm HK}\nolimits^{\bullet}(A)$-bimodule in $A\text{-{Bimod}}$, so that $H(\tilde{\varphi}):\mathop{\rm HK}\nolimits_{\bullet}(A,A\stackrel{{\scriptstyle o}}{{\otimes}}A)\rightarrow H(K(A))$ is an isomorphism of graded $\mathop{\rm HK}\nolimits^{\bullet}(A)$-bimodules in $A\text{-{Bimod}}$.

Let us give explicitly the underlying $\tilde{A}$-bimodule structure of the DG $\tilde{A}$-bimodule $K(A)$. Consider $z=(\alpha\otimes\beta)\otimes_{k^{e}}x_{1}\ldots x_{q}$ in $(A\stackrel{{\scriptstyle o}}{{\otimes}}A)\otimes_{k^{e}}W_{q}$ and $f$ in $\mathop{\rm Hom}\nolimits_{k^{e}}(W_{p},A)$, we easily derive from (7) that the left action of $f$ on $K(A)$ is defined by

[TABLE]

Analogously, using (8), we define the right action of $f$ on $K(A)$ by

[TABLE]

The fundamental formula (13) reduces to

[TABLE]

on $K(A)$, where $z^{\prime}=\beta\otimes_{k}x_{1}\ldots x_{q}\otimes_{k}\alpha$, and

[TABLE]

The differential $\mathfrak{e}_{A}\underset{K}{\frown}-$ induces a differential, still denoted by $\partial_{\frown}$, on $H(K(A))$. The homology of $(H(K(A)),\partial_{\frown})$ is denoted by $H^{hi}(K(A))$ and is called the higher homology of $K(A)$. Then $H^{hi}(K(A))$ is a graded $\mathop{\rm HK}\nolimits_{hi}^{\bullet}(A)$-bimodule in $A\text{-{Bimod}}$ and $H(H(\tilde{\varphi})):\mathop{\rm HK}\nolimits^{hi}_{\bullet}(A,A\stackrel{{\scriptstyle o}}{{\otimes}}A)\rightarrow H^{hi}(K(A))$ is an isomorphism of graded $\mathop{\rm HK}\nolimits_{hi}^{\bullet}(A)$-bimodules in $A\text{-{Bimod}}$.

4 Poincaré Van den Bergh duality of preprojective algebras

4.1 Preprojective algebras

Throughout this section, $\Delta$ is a connected graph whose vertex set and edge set are finite. Following a usual presupposition in the papers devoted to Hochschild (co)homology of preprojective algebras, we assume that the graph $\Delta$ is not labelled, that is, the labels of the edges are all equal to $(1,1)$ [4, Definition 4.1.9]. In particular, the Dynkin graphs are limited to types ADE, and the Euclidean (or extended) Dynkin graphs are limited to types $\tilde{\textup{A}}\tilde{\textup{D}}\tilde{\textup{E}}$ [4, Definition 4.5.1].

Let $Q$ be a quiver whose underlying graph is $\Delta$. Define a quiver $Q^{\ast}$ whose vertex set is $Q_{0}$ and whose arrow set is $Q_{1}^{\ast}=\{a^{\ast};a\in Q_{1}\}$ where $\mathfrak{s}(a^{\ast})=\mathfrak{t}(a)$ and $\mathfrak{t}(a^{\ast})=\mathfrak{s}(a)$. Let $\mkern 2.0mu\overline{\mkern-2.0muQ\mkern 1.0mu}\mkern-1.0mu$ be the double quiver of $Q$, that is, the quiver whose vertex set is $\overline{Q}_{0}=Q_{0}$ and whose arrow set is the disjoint union $\overline{Q}_{1}=Q_{1}\cup Q_{1}^{\ast}$. We shall view $(-)^{\ast}$ as an involution of $\overline{Q}_{1}$.

Let $\mathbb{F}$ be a field. As before, we denote the ring $\mathbb{F}Q_{0}$ by $k$ and the $k$-bimodule $\mathbb{F}\overline{Q}_{1}$ by $V$ and we identify the graded $k$-algebras $T_{k}(V)\cong\mathbb{F}\overline{Q}$ (see Subsection 2.1).

The preprojective algebra associated with the graph $\Delta$ over the field $\mathbb{F}$ is the quadratic $k$-algebra $A(\Delta)$ over $\overline{Q}$ defined by $A(\Delta)=\mathbb{F}\overline{Q}/(R)$, where the sub-$k$-bimodule $R$ of $\mathbb{F}\overline{Q}_{2}$ is generated by

[TABLE]

where $\varepsilon(a)=1$ if $a\in Q_{1}$, $\varepsilon(a)=-1$ if $a\in Q_{1}^{\ast}$.

If $Q^{\prime}$ is another quiver whose underlying graph is $\Delta$, and $R^{\prime}$ is the sub-$k$-bimodule of $\mathbb{F}\overline{Q^{\prime}}_{2}$ generated by the relations $\sigma_{i}^{\prime}=\sum_{\begin{subarray}{c}a\in\mkern 2.0mu\overline{\mkern-2.0muQ\mkern 1.0mu}\mkern-1.0mu^{\prime}_{1}\\ \mathfrak{t}(a)=i\end{subarray}}\varepsilon(a)aa^{*}$ for all $i\in Q_{0}^{\prime}=Q_{0}$, then the preprojective algebras $\mathbb{F}\overline{Q}/(R)$ and $\mathbb{F}\overline{Q^{\prime}}/(R^{\prime})$ are isomorphic, the isomorphism being given by exchanging pairs of arrows $a$ and $a^{*}$ and changing the sign of one arrow in each pair (see [13, Remark 2.2(3)], or [38, Lemma 1.3.7] for a complete proof in the case of generalised preprojective algebras). Therefore, according to Subsection 2.7, the quadratic $k$-algebra $A(\Delta)$ and the (general, higher) Koszul calculus of $A(\Delta)$ depend only on the graph $\Delta$ and not on $Q$, justifying the notation $A(\Delta)$. If $\Delta$ is a tree, $A(\Delta)$ is isomorphic to the preprojective algebra defined without signs (that is, $\varepsilon(a)=1$ for all $a\in\mkern 2.0mu\overline{\mkern-2.0muQ\mkern 1.0mu}\mkern-1.0mu_{1}$, as in [18, 19, 20]).

If $\Delta=\mathrm{A}_{1}$, then $A(\Delta)=k$. If $\Delta=\mathrm{A}_{2}$, then $R=\mathbb{F}\overline{Q}_{2}$ and $A(\Delta)=\mathbb{F}Q_{0}\oplus\mathbb{F}\overline{Q}_{1}$. These quadratic $k$-algebras are Koszul, but they are the only exceptions among the Dynkin graphs. More precisely, the following standard result holds, for which we just give proof references (see also [10, Corollary 4.3]).

**Proposition **\theDf

Assume that the graph $\Delta$ is distinct from $\mathrm{A}_{1}$ and $\mathrm{A}_{2}$. The following are equivalent.

1. \edefitit(i)

$\Delta$* is Dynkin of type ADE.* 2. \edefitit(ii)

$A(\Delta)$* is not Koszul.* 3. \edefitit(iii)

$A(\Delta)$* is finite dimensional.*

Proof 4.1**.**

The equivalence (i)$\Leftrightarrow$(ii) is treated in [33] if $\Delta$ is a tree, in [30] otherwise. The equivalence (i)$\Leftrightarrow$(iii) for any Dynkin graph is cited in [34] as a result by Gelfand and Ponomarev [27].

Sections 2 and 3 can be applied to preprojective algebras. For example, according to the remark following Lemma 2.4, the fundamental 1-cocycle $\mathfrak{e}_{A(\Delta)}$ is not a coboundary if $\Delta$ has a loop or if $\mathop{\rm char}\nolimits\mathbb{F}\neq 2$ and $\Delta\neq\mathrm{A}_{1}$. In the remainder of this section, we often abbreviate $A(\Delta)$ to $A$ and we freely use notations and results from Sections 2 and 3.

4.2 The Koszul complex $K(A)$ has length 2

If $\Delta=\mathrm{A}_{1}$, then $K(A)$ has length 0. If $\Delta=\mathrm{A}_{2}$, then $K(A)$ has infinite length. However, when $\Delta$ is not Dynkin ADE, the algebra $A=A(\Delta)$ has global dimension $2$ (this is a consequence of [10, Proposition 4.2], inspired by manuscript notes of Crawley-Boevey), and since $k\cong\mathbb{F}^{|Q_{0}|}$ is separable, it follows that the minimal projective $A$-bimodule resolution of $A$, which is $K(A)$ because $A$ is Koszul, has length $2$ (see for instance [37, Proposition 3.18]).

Actually, the fact that the length of $K(A)$ is $2$ is true for all graphs $\Delta$ other than $\mathrm{A}_{1}$ and $\mathrm{A}_{2}$, and we now give a unified proof of this.

**Theorem **\theDf

Let $A=A(\Delta)$ be a preprojective algebra over $\mathbb{F}$ with $\Delta\neq\mathrm{A}_{1}$ and $\Delta\neq\mathrm{A}_{2}$. Then the Koszul complex $K(A)$ of $A$ has length 2. Consequently, $\mathop{\rm HK}\nolimits^{p}(A,M)\cong\mathop{\rm HK}\nolimits_{p}(A,M)=0$ for all $A$-bimodules $M$ and all $p\geqslant 3$.

Proof 4.2**.**

From the defining equality (2) of $W_{p}$, we have $W_{p}=(W_{p-1}\otimes_{k}V)\cap(V\otimes_{k}W_{p-1})$ for all $p\geqslant 3$. Moreover $R\neq 0$, therefore it is enough to prove that $W_{3}=0,$ that is, $(R\otimes_{k}V)\cap(V\otimes_{k}R)=0$. For that, we only assume that $\Delta\neq\mathrm{A}_{1}$. Our goal is to prove that $W_{3}\neq 0$ implies $\Delta=\mathrm{A}_{2}$.

Let $u$ be a non-zero element in $W_{3}$, viewed as an element in $\mathbb{F}\mkern 2.0mu\overline{\mkern-2.0muQ\mkern 1.0mu}\mkern-1.0mu_{3}.$ There exist vertices $e,f$ in $Q_{0}$ such that $euf\neq 0$, therefore we may assume that $u$ is in $eW_{3}f.$ Then $u$ can be written uniquely as

[TABLE]

We now use the fact that $\mkern 2.0mu\overline{\mkern-2.0muQ\mkern 1.0mu}\mkern-1.0mu_{1}$ is the disjoint union of $Q_{1}$ and $Q_{1}^{*}$ and the definition of $\varepsilon$ to write

[TABLE]

From these expressions, we obtain the following identities in the path algebra $\mathbb{F}\mkern 2.0mu\overline{\mkern-2.0muQ\mkern 1.0mu}\mkern-1.0mu$:

[TABLE]

Indeed, identity (1) follows from the fact that no other path that occurs in the expressions of $u$ ends with two arrows in $Q^{*}$, and the other identities are obtained from similar arguments.

In the path algebra $\mathbb{F}\mkern 2.0mu\overline{\mkern-2.0muQ\mkern 1.0mu}\mkern-1.0mu$, where there are no relations between paths, the identities (1) to (4) above are equivalent to

[TABLE]

We have assumed that $u\neq 0$, so that either there exists $\alpha\in eQ_{1}f$ such that $\lambda_{\alpha}\neq 0$ or there exists $\alpha\in fQ_{1}e$ such that $\lambda_{\alpha^{*}}\neq 0,$ using the first expression of $u.$ We separate the two cases.

Assume that there exists $\alpha\in eQ_{1}f$ such that $\lambda_{\alpha}\neq 0$. Then it follows from identity (8) that $Q_{1}e$ is empty. From (5), for all $a\in eQ_{1}$, there exist $\beta\in eQ_{1}f$ and $b\in Q_{1}f$ such that $\lambda_{\alpha}aa^{*}\alpha=-\mu_{\beta}\beta b^{*}b.$ Hence $\beta=a=b=\alpha$ and therefore $eQ_{1}=\left\{\alpha\right\}=eQ_{1}f$ and $\mu_{\beta}=-\lambda_{\alpha}\neq 0.$ From (9), it follows that $fQ_{1}$ is empty. Finally, (5) becomes

[TABLE]

so that $\sum_{\begin{subarray}{c}b\in Q_{1}f\\ b\neq\alpha\end{subarray}}\alpha b^{*}b=0$ and hence $Q_{1}f=\left\{\alpha\right\}.$

We have proved that $Q_{1}e=\emptyset=fQ_{1}$ so that in particular $e\neq f$, and that $Q_{1}f=\left\{\alpha\right\}=eQ_{1}=eQ_{1}f$. Finally, $Q=e\stackrel{{\scriptstyle\alpha}}{{\leftarrow}}f$ and $\Delta=A_{2}$.

In the case where there exists $\alpha\in fQ_{1}e$ such that $\lambda_{\alpha^{*}}\neq 0$, a similar proof using (7), (10) and (6) shows that $Q=f\stackrel{{\scriptstyle\alpha}}{{\leftarrow}}e$ and $\Delta=A_{2}$.

As an immediate consequence of Proposition 2.4 and Theorem 4.2, we obtain that in the Koszul calculus of $A(\Delta)$, the Koszul cup product is graded commutative and the Koszul cap product is graded symmetric. The precise statement is the following.

**Corollary **\theDf

Let $A=A(\Delta)$ be a preprojective algebra over $\mathbb{F}$ with $\Delta\neq\mathrm{A}_{1}$ and $\Delta\neq\mathrm{A}_{2}$. We consider an $A$-bimodule $M$. For any $\alpha\in\mathop{\rm HK}\nolimits^{\bullet}(A,M)$, $\beta\in\mathop{\rm HK}\nolimits^{\bullet}(A)$ and $\gamma\in\mathop{\rm HK}\nolimits_{\bullet}(A)$, we have the identities

[TABLE]

The same conclusion holds if $\Delta=\mathrm{A}_{1}$ (obvious) and if $\Delta=\mathrm{A}_{2}$ (because $A$ is Koszul).

4.3 Duality in Koszul (co)homology of preprojective algebras

There is a remarkable duality between Koszul homology and cohomology for preprojective algebras. This duality is realised as a cap action by a Koszul 2-chain $\omega_{0}\in A\otimes_{k^{e}}R$ defined for any graph $\Delta$ by

[TABLE]

From $\sigma_{i}=\sum_{a\in\mkern 2.0mu\overline{\mkern-2.0muQ\mkern 1.0mu}\mkern-1.0mu_{1},\,\mathfrak{t}(a)=i}\varepsilon(a)aa^{*}$, we get

[TABLE]

Then it is easy to check that $\omega_{0}$ is a Koszul 2-cycle. Being homogeneous of weight 0, $\omega_{0}$ is not a 2-boundary whenever $\Delta\neq\mathrm{A}_{1}$.

In the following statement, we need the DG algebra $\tilde{A}$ and the DG $\tilde{A}$-bimodules $\mathop{\rm Hom}\nolimits_{k^{e}}(W_{\bullet},M)$ and $M\otimes_{k^{e}}W_{\bullet}$, introduced just before Definition 2.3.

**Theorem **\theDf

Let $A=A(\Delta)$ be a preprojective algebra over $\mathbb{F}$ with $\Delta\neq\mathrm{A}_{1}$ and $\Delta\neq\mathrm{A}_{2}$. Consider the Koszul 2-cycle $\omega_{0}=\sum_{i\in Q_{0}}e_{i}\otimes\sigma_{i}\in A\otimes_{k^{e}}R$. For each Koszul $p$-cochain $f$ with coefficients in an $A$-bimodule $M$, we define the Koszul $(2-p)$-chain $\theta_{M}(f)$ with coefficients in $M$ by

[TABLE]

Then the equalities

[TABLE]

hold for any Koszul cochains $f$ and $g$ with coefficients in $A$-bimodules $M$ and $N$ respectively.

Moreover the linear map $\theta_{M}:\mathop{\rm Hom}\nolimits_{k^{e}}(W_{\bullet},M)\rightarrow M\otimes_{k^{e}}W_{2-\bullet}$ is an isomorphism of DG $\tilde{A}$-bimodules.

It follows that $H(\theta_{M}):\mathop{\rm HK}\nolimits^{\bullet}(A,M)\rightarrow\mathop{\rm HK}\nolimits_{2-\bullet}(A,M)$ is an isomorphism of graded $\mathop{\rm HK}\nolimits^{\bullet}(A)$-bimodules and that $H(H(\theta_{M})):\mathop{\rm HK}\nolimits_{hi}^{\bullet}(A,M)\rightarrow\mathop{\rm HK}\nolimits^{hi}_{2-\bullet}(A,M)$ is an isomorphism of graded $\mathop{\rm HK}\nolimits_{hi}^{\bullet}(A)$-bimodules.

Proof 4.3**.**

First we show that $f\underset{K}{\frown}\omega_{0}=\omega_{0}\underset{K}{\frown}f$ for all $f\in\mathop{\rm Hom}\nolimits_{k^{e}}(W_{p},M)$. Using the definition of $\omega_{0}$ and the equalities (13), (7) and (8), we obtain for $p=0,1,2,$

[TABLE]

Next, for $f\in\mathop{\rm Hom}\nolimits_{k^{e}}(W_{p},M)$ and $g\in\mathop{\rm Hom}\nolimits_{k^{e}}(W_{q},M)$, we have

[TABLE]

providing equalities (15). Therefore $\theta_{M}:\mathop{\rm Hom}\nolimits_{k^{e}}(W_{\bullet},M)\rightarrow M\otimes_{k^{e}}W_{2-\bullet}$ is a morphism of graded $\tilde{A}$-bimodules, where $\tilde{A}$ is just considered as a graded algebra. It remains to examine what happens for the Koszul differentials.

Assuming $M=A$ in the equalities (15), we derive

[TABLE]

Combining $\theta_{N}([\mathfrak{e}_{A},g]_{\underset{K}{\smile}})=[\mathfrak{e}_{A},\theta_{N}(g)]_{\underset{K}{\frown}}$ with $b_{K}=-[\mathfrak{e}_{A},-]_{\underset{K}{\smile}}$ and $b^{K}=-[\mathfrak{e}_{A},-]_{\underset{K}{\frown}}$, we deduce that $\theta_{M}$ is a morphism of complexes, thus a morphism of DG $\tilde{A}$-bimodules.

We prove that $\theta_{M}$ is an isomorphism by giving an inverse map $\eta:M\otimes_{k^{e}}W_{2-\bullet}\rightarrow\mathop{\rm Hom}\nolimits_{k^{e}}(W_{\bullet},M)$. We define $\eta_{p}:M\otimes_{k^{e}}W_{2-p}\rightarrow\mathop{\rm Hom}\nolimits_{k^{e}}(W_{p},M)$ for $p=0,1,2,$ by

[TABLE]

where $\delta$ is the Kronecker symbol. It is routine to verify that these linear maps are well-defined and form an inverse map for $\theta_{M}$.

Finally the isomorphism $H(\theta_{M})$ of graded $\mathop{\rm HK}\nolimits^{\bullet}(A)$-bimodules satisfies

[TABLE]

for all $\alpha\in\mathop{\rm HK}\nolimits^{\bullet}(A,M)$. Therefore $H(\theta_{M})$ is a morphism of complexes for higher (co)homologies. Taking higher (co)homologies, we get a $\mathop{\rm HK}\nolimits^{\bullet}_{hi}(A)$-bimodule isomorphism

[TABLE]

By analogy with the Poincaré duality in singular (co)homology [31] and with the Van den Bergh duality in Hochschild (co)homology [41, 32], we say that the isomorphism

[TABLE]

is a Poincaré Van den Bergh duality for Koszul (co)homology, of fundamental class $\overline{\omega}_{0}$, where $\overline{\omega}_{0}\in\mathop{\rm HK}\nolimits_{2}(A)_{0}$. In the next subsection, we extract from this duality a generalisation of the 2-Calabi-Yau property.

Unless $\Delta$ has no loop and $\mathop{\rm char}\nolimits\mathbb{F}=2$, the class $\overline{\mathfrak{e}}_{A}\in\mathop{\rm HK}\nolimits^{1}(A)_{1}$ is non-zero, hence

[TABLE]

is non-zero in $\mathop{\rm HK}\nolimits_{1}(A)_{1}$. Consequently, the fundamental class $\overline{\omega}_{0}$ of the Poincaré Van den Bergh duality is not a cycle for the higher Koszul homology, so that the isomorphism $H(H(\theta_{M}))$ cannot be naturally expressed as a cap action.

The class $H(\theta_{A})(\overline{\mathfrak{e}}_{A})$ is the class of the Koszul 1-cycle $\omega_{0}\underset{K}{\frown}\mathfrak{e}_{A}$ where

[TABLE]

It is interesting to view the last element as the image by the canonical linear map $can:V\otimes_{k}V\rightarrow V\otimes_{k^{e}}V$ of the element

[TABLE]

In the identification $V\otimes_{k}V\cong\mathbb{F}\mkern 2.0mu\overline{\mkern-2.0muQ\mkern 1.0mu}\mkern-1.0mu_{2}$, $V\otimes_{k^{e}}V$ is identified with the subspace of cycles of length 2 and the map $can$ is identified with the projection whose kernel is the space spanned by the non-cyclic paths. Since $R$ is generated by the cycles $\sigma_{i}$, we can make the identification $\omega_{0}\underset{K}{\frown}\mathfrak{e}_{A}=w$. The element $w$ was defined in [14, Proposition 8.1.1] as a representative of a bi-symplectic 2-form $\omega$. Bi-symplectic 2-forms were introduced by Crawley-Boevey, Etingof and Ginzburg as an essential ingredient of the Hamiltonian reduction in noncommutative geometry [14]; they are related to the double Poisson algebras defined by Van den Bergh [42].

{Rm}

Assume that $\Delta=\mathrm{A}_{2}$, so that $A$ is defined by the quiver

[TABLE]

subject to the relations $\sigma_{0}=-a^{*}a$, $\sigma_{1}=aa^{*}$. Then the statement of Theorem 4.3 is valid in a weaker form, namely the isomorphisms involved are only morphisms. Moreover, $\theta_{M}$ is bijective only in degree $q$, $0\leq q\leq 2$, with the same inverse $\eta_{q}$. More generally, for any $p\geq 1$, the Koszul ($2p$)-cycle

[TABLE]

provides a morphism $\omega_{p-1}\stackrel{{\scriptstyle[}}{{K}}]{}{\frown}-$ which is bijective only in degree $q$, $0\leq q\leq 2p$. From that, we deduce an isomorphism $\overline{\omega}_{p-1}\stackrel{{\scriptstyle[}}{{K}}]{}{\frown}-$ from $\mathop{\rm HK}\nolimits^{q}(A,M)$ to $\mathop{\rm HK}\nolimits_{2p-q}(A,M)$ for $0<q<2p$. Varying $p$, we obtain the following duality and $2$-periodicity

[TABLE]

Using this for $M=A$, it is straightforward to compute explicitly the restricted Koszul calculus of $A$. We leave the details to the reader. Notice that, since $A$ is Koszul, we have the same duality and $2$-periodicity for Hochschild (co)homology, recovering a known result as a consequence of remarkable isomorphisms due to Eu and Schedler [24, Theorem 2.3.27, Theorem 2.3.47], here applied to [24, Example 2.3.10, Corollary 2.1.13].

4.4 Deriving an adapted 2-Calabi-Yau property

Let $A=A(\Delta)$ be a preprojective algebra over $\mathbb{F}$ with $\Delta\neq\mathrm{A}_{1}$ and $\Delta\neq\mathrm{A}_{2}$. Let $M$ be an $A$-bimodule. Assume that $B$ is a unital associative algebra such that $M$ is a right $B$-module compatible with the $A$-bimodule structure (see Subsection 3.1). Denote by $\text{{Mod}-}B$ the category of right $B$-modules. Recall that $\tilde{A}$ denotes the DG algebra $(\mathop{\rm Hom}\nolimits_{k^{e}}(W_{\bullet},A),b_{K},\underset{K}{\smile})$.

According to Subsection 3.2, $M\otimes_{k^{e}}W_{\bullet}$ and $\mathop{\rm Hom}\nolimits_{k^{e}}(W_{\bullet},M)$ are DG $\tilde{A}$-bimodules in $\text{{Mod}-}B$. Moreover, $\mathop{\rm HK}\nolimits_{\bullet}(A,M)$ and $\mathop{\rm HK}\nolimits^{\bullet}(A,M)$ are graded $\mathop{\rm HK}\nolimits^{\bullet}(A)$-bimodules in $\text{{Mod}-}B$. Finally, $\mathop{\rm HK}\nolimits^{hi}_{\bullet}(A,M)$ and $\mathop{\rm HK}\nolimits_{hi}^{\bullet}(A,M)$ are graded $\mathop{\rm HK}\nolimits_{hi}^{\bullet}(A)$-bimodules in $\text{{Mod}-}B$.

**Lemma **\theDf

The map $\theta_{M}:\mathop{\rm Hom}\nolimits_{k^{e}}(W_{\bullet},M)\rightarrow M\otimes_{k^{e}}W_{2-\bullet}$ is an isomorphism of DG $\tilde{A}$-bimodules in $\text{{Mod}-}B$. Moreover, $H(\theta_{M}):\mathop{\rm HK}\nolimits^{\bullet}(A,M)\rightarrow\mathop{\rm HK}\nolimits_{2-\bullet}(A,M)$ is an isomorphism of graded $\mathop{\rm HK}\nolimits^{\bullet}(A)$-bimodules in $\text{{Mod}-}B$, and $H(H(\theta_{M})):\mathop{\rm HK}\nolimits_{hi}^{\bullet}(A,M)\rightarrow\mathop{\rm HK}\nolimits^{hi}_{2-\bullet}(A,M)$ is an isomorphism of graded $\mathop{\rm HK}\nolimits_{hi}^{\bullet}(A)$-bimodules in $\text{{Mod}-}B$.

Proof 4.4**.**

It is enough to prove that $\theta_{M}:f\mapsto\omega_{0}\underset{K}{\frown}f$ is $B$-linear. For a $k$-bimodule morphism $f:W_{p}\rightarrow M$, $z=a\otimes_{k^{e}}x_{1}\ldots x_{q}\in A\otimes_{k^{e}}W_{q}$ and $b\in B$, we verify the identities

[TABLE]

The first one uses the fact that the right actions of $A$ and $B$ on $M$ commute, while the second one uses the fact that $M$ is an $A$-$B$-bimodule (see (iii) in Lemma 3.1). Applying the second one to $z=\omega_{0}$, we obtain that $\theta_{M}$ is $B$-linear.

We specialise this lemma to $M=B=A^{e}$ and, using the isomorphism $\tilde{\varphi}$ in Subsection 3.3, we identify $A^{e}\otimes_{k^{e}}W_{\bullet}$ with $K(A)$ to get the next proposition.

**Proposition **\theDf

Let $A=A(\Delta)$ be a preprojective algebra over $\mathbb{F}$ with $\Delta\neq\mathrm{A}_{1}$ and $\Delta\neq\mathrm{A}_{2}$. The map

[TABLE]

is an isomorphism of DG $\tilde{A}$-bimodules in $A\text{-{Bimod}}$. Moreover,

[TABLE]

is an isomorphism of graded $\mathop{\rm HK}\nolimits^{\bullet}(A)$-bimodules in $A\text{-{Bimod}}$.

The homology of $K(A)$ is isomorphic to $A$ in degree 0, and to [math] in degree 1, hence we obtain a generalisation of the 2-Calabi-Yau property, formulated as follows.

**Theorem **\theDf

Let $A=A(\Delta)$ be a preprojective algebra over $\mathbb{F}$ with $\Delta\neq\mathrm{A}_{1}$ and $\Delta\neq\mathrm{A}_{2}$. Then the $\mathop{\rm HK}\nolimits^{\bullet}(A)^{e}$-$A^{e}$-bimodules $\mathop{\rm HK}\nolimits^{\bullet}(A,A^{e})$ and $H_{2-\bullet}(K(A))$ are isomorphic. In particular, we have the following.

1. \edefitit(i)

The $A$-bimodule $\mathop{\rm HK}\nolimits^{2}(A,A^{e})$ is isomorphic to the $A$-bimodule $A$. 2. \edefitit(ii)

$\mathop{\rm HK}\nolimits^{1}(A,A^{e})=0$. 3. \edefitit(iii)

The $A$-bimodule $\mathop{\rm HK}\nolimits^{0}(A,A^{e})$ is isomorphic to the $A$-bimodule $H_{2}(K(A))$.

Since $H_{1}(K(A))\cong\mathop{\rm HK}\nolimits^{1}(A,A^{e})=0$, the higher Koszul differentials vanish. Therefore $H^{hi}_{p}(K(A))\cong H_{p}(K(A))$, $\mathop{\rm HK}\nolimits_{hi}^{p}(A,A^{e})\cong\mathop{\rm HK}\nolimits^{p}(A,A^{e})$ and $H(H(\theta_{A^{e}}))\cong H(\theta_{A^{e}})$.

From the generator $1\otimes_{k}1$ of the $A$-bimodule $H_{0}(K(A))$, we draw from (i) a generator of the free $A$-bimodule $\mathop{\rm HK}\nolimits^{2}(A,A^{e})$ defined as the class of $f:R\rightarrow A\stackrel{{\scriptstyle o}}{{\otimes}}A$ with $f(\sigma_{i})=e_{i}\otimes e_{i}$ for any $i$.

In (iii), the $A$-bimodules are never [math] when $\Delta$ is Dynkin of types $\mathrm{ADE}$ since $A$ is not Koszul in this case. This situation is drastically different from the 2-Calabi-Yau property defined by Ginzburg in terms of the Hochschild cohomology spaces $\mathop{\rm HH}\nolimits^{p}(A,A^{e})$ [28, §3.2]. In Ginzburg’s definition, $\mathop{\rm HH}\nolimits^{p}(A,A^{e})=0$ for all $p<2$.

5 Generalisations of Calabi-Yau algebras

5.1 Duality for Koszul complex Calabi-Yau algebras

From Theorem 4.4, we are led to introduce a general definition in the framework of quiver algebras with homogeneous quadratic relations (see Section 2). The notation introduced in Section 2 stands throughout. We are interested in quadratic $k$-algebras $A=T_{k}(V)/(R)$ over a finite quiver $\mathcal{Q}$ as defined in Subsection 2.1, and in the Koszul calculus of $A$ as presented in the remainder of Section 2. Note that $\mathcal{Q}_{1}=\overline{Q}_{1}$ if we want to specialise to preprojective algebras.

For the definition of the bounded derived category $\mathcal{D}^{b}(\mathcal{C})$ of an abelian category $\mathcal{C}$, we refer to [44, Chapter 10]. Recall that $A\text{-{Bimod}}$ denotes the category of $A$-bimodules.

{Df}

Let $A=T_{k}(V)/(R)$ be a quadratic $k$-algebra over $\mathcal{Q}$. Let $n\geqslant 0$ be an integer. We say that $A$ is Koszul complex Calabi-Yau (Kc-Calabi-Yau) of dimension $n$, or $n$-Kc-Calabi-Yau, if

1. \edefnit(i)

the Koszul bimodule complex $K(A)$ of $A$ has length $n$, and 2. \edefnit(ii)

$\mathop{\rm RHom}\nolimits_{A^{e}}(K(A),A^{e})\cong K(A)[-n]$ in $\mathcal{D}^{b}(A\text{-{Bimod}})$.

Property (ii) is equivalent to saying that there is an $A$-bimodule quasi-isomorphism from $\mathop{\rm Hom}\nolimits_{k^{e}}(W_{\bullet},A^{e})$ to $K(A[-n]$. According to Theorem 4.2 and Proposition 4.4, a preprojective algebra $A(\Delta)$ over $\mathbb{F}$ with $\Delta\neq\mathrm{A}_{1}$ and $\Delta\neq\mathrm{A}_{2}$ is Kc-Calabi-Yau of dimension $2$. In fact, the isomorphism $\theta_{A^{e}}$ induces an isomorphism $\mathop{\rm RHom}\nolimits_{A^{e}}(K(A),A^{e})\rightarrow K(A)[-2]$ in $\mathcal{D}^{b}(A\text{-{Bimod}})$.

Let us recall Ginzburg’s definition of Calabi-Yau algebras [28, Definition 3.2.3] as reformutated by Van den Bergh [43, Definition 8.2]. We shall apply this definition to quadratic quiver algebras by considering it as $\mathbb{F}$-algebras.

{Df}

An associative $\mathbb{F}$-algebra $A$ is said to be Calabi-Yau of dimension $n$ if

(i) $A$ is homologically smooth, that is, $A$ has a bounded resolution by finitely generated projective $A$-bimodules,

(ii) $\mathop{\rm RHom}\nolimits_{A^{e}}(A,A^{e})\cong A[-n]$ in $\mathcal{D}^{b}(A\text{-{Bimod}})$.

Definition 5.1 is a true generalisation of Definition 5.1 for quadratic quiver algebras. If $\Delta$ is Dynkin of type ADE, $A(\Delta)$ is not Calabi-Yau in Ginzburg’s definition since $A(\Delta)$ is not homologically smooth in this case (the minimal projective resolution of $A(\Delta)$ has infinite length). However, the two definitions coincide if $A$ is Koszul.

**Proposition **\theDf

Let $A=T_{k}(V)/(R)$ be a quadratic $k$-algebra over $\mathcal{Q}$. Assume that $A$ is Koszul. Then $A$ is $n$-Kc-Calabi-Yau if and only if $A$ is $n$-Calabi-Yau.

Proof 5.1**.**

Assume that $A$ is $n$-Kc-Calabi-Yau. Property (i) and the fact that $A$ is Koszul show that $A$ is homologically smooth. Furthermore, $K(A)\cong A$ in $\mathcal{D}^{b}(A\text{-{Bimod}})$. Thus $\mathop{\rm RHom}\nolimits_{A^{e}}(A,A^{e})\cong A[-n]$ in $\mathcal{D}^{b}(A\text{-{Bimod}})$, and we recover Definition 5.1.

Assume that $A$ is $n$-Calabi-Yau. We know that $n$ is equal to the projective dimension of the $A$-bimodule $A$ [41] which in turn is equal to the length of a minimal projective resolution of $A$ (see for instance [5]). Hence $K(A)$ has length $n$ and $K(A)\cong A$ in $\mathcal{D}^{b}(A\text{-{Bimod}})$, which allows us to conclude that $A$ is $n$-Kc-Calabi-Yau.

If the graph $\Delta$ is not Dynkin ADE, we know that $A(\Delta)$ is Koszul (Proposition 4.1), thus we recover the fact that $A(\Delta)$ is 2-Calabi-Yau [14, 9].

In Subsection 2.8, we have seen that $K(A)$ and the minimal projective resolution $P(A)$ coincide up to the homological degree 2. Therefore, if $n\in\left\{0,1\right\}$ and if $A$ is $n$-Calabi Yau or $n$-Kc-Calabi-Yau, then $P(A)=K(A)$ so that $A$ is Koszul, and it follows that the two definitions are equivalent when $n\in\{0,1\}$.

**Proposition **\theDf

Let $A=T_{k}(V)/(R)$ be a quadratic $k$-algebra over $\mathcal{Q}$. Then $A$ is Calabi-Yau of dimension 0 if and only if $\mathcal{Q}_{1}=\emptyset$.

We leave the proof as an exercise. If $A$ is Calabi-Yau of dimension 1, then $R=0$, that is, $A=T_{k}(V)\cong\mathbb{F}\mathcal{Q}$ with $\mathcal{Q}_{1}\neq\emptyset$. It is indeed 1-Calabi-Yau if $\mathcal{Q}$ has only one vertex and one loop, but we have not yet found other examples when $\mathcal{Q}$ is connected.

If $A$ is $n$-Calabi-Yau, the Van den Bergh duality theorem states that the vector spaces $\mathop{\rm HH}\nolimits^{p}(A,M)$ and $\mathop{\rm HH}\nolimits_{n-p}(A,M)$ are isomorphic [41]. From Definition 5.1, we draw an analogous duality theorem for Koszul homology/cohomology.

**Theorem **\theDf

Let $A$ be a Koszul complex Calabi-Yau algebra of dimension $n$ over $\mathcal{Q}$. Then for any $A$-bimodule $M$, the vector spaces $\mathop{\rm HK}\nolimits^{p}(A,M)$ and $\mathop{\rm HK}\nolimits_{n-p}(A,M)$ are isomorphic.

Proof 5.2**.**

Denote by VectF the category of $\mathbb{F}$-vector spaces. For any $A$-bimodule $M$, the left derived functor $M\stackrel{{\scriptstyle L}}{{\otimes}}_{A^{e}}-$ and the right derived functor $\mathop{\rm RHom}\nolimits_{A^{e}}(-,M)$ are defined from $\mathcal{D}^{b}(A\text{-{Bimod}})$ to \mathcal{D}^{b}(\text{{Vect}{}_{\mathbb{F}}}) [44, Chapter 10].

Our proof is based on a natural transformation depending on an $A$-bimodule $M$. Let F:A\text{-{Bimod}}\rightarrow\text{{Vect}{}{\mathbb{F}}} be the functor $F:N\mapsto\mathop{\rm Hom}\nolimits_{A^{e}}(N,M)$ where $M$ and $N$ are seen as right $A^{e}$-modules. Specialising to $M=A^{e}$ in $F$, we define a functor $G:A\text{-{Bimod}}\rightarrow A\text{-{Bimod}}$. Let H:A\text{-{Bimod}}\rightarrow\text{{Vect}{}{\mathbb{F}}} be the functor $H:N^{\prime}\mapsto M\otimes_{A^{e}}N^{\prime}$ where $N^{\prime}$ is viewed as a left $A^{e}$-module. Then we define a linear map

[TABLE]

by $\phi_{M}(m\otimes_{A^{e}}g)(x)=m.g(x)$ for $m\in M$, $g\in\mathop{\rm Hom}\nolimits_{A^{e}}(N,A^{e})$ and $x\in N$. This map is functorial in $N$, defining a natural transformation $\phi_{M}:H\circ G\Rightarrow F$.

If the $A$-bimodule $P$ is projective and finitely generated,

[TABLE]

is an isomorphism. It is standard, see e.g. [11, Proposition (8.3) (c)]. Then for any bounded chain complex $C$ of finitely generated projective $A$-bimodules, $\phi_{M}$ induces in \mathcal{D}^{b}(\text{{Vect}{}_{\mathbb{F}}}) an isomorphism

[TABLE]

Applying it to $C=K(A)$ and using (ii) in Definition 5.1, we get an isomorphism

[TABLE]

in \mathcal{D}^{b}(\text{{Vect}{}_{\mathbb{F}}}). Taking homology, we deduce that $\mathop{\rm HK}\nolimits^{p}(A,M)\cong\mathop{\rm HK}\nolimits_{n-p}(A,M)$ as vector spaces.

5.2 Koszul complex Calabi-Yau algebras versus Calabi-Yau algebras

Recall that if $\Delta\neq\mathrm{A}_{1}$ and $\Delta\neq\mathrm{A}_{2}$, then $A(\Delta)$ is $2$-Kc-Calabi-Yau. But observe that if $A(\Delta)$ is moreover $2$-Calabi-Yau, then $A(\Delta)$ is Koszul. We are led to the following conjecture. {Cn} Let $A=T_{k}(V)/(R)$ be a quadratic $k$-algebra over $\mathcal{Q}$. If $A$ is $n$-Calabi-Yau and $n$-Kc-Calabi-Yau, then $A$ is Koszul. In other words, if $A$ is not Koszul, the properties $n$-Calabi-Yau and $n$-Kc-Calabi-Yau are not simultaneously true.

**Proposition **\theDf

Let $A=T_{k}(V)/(R)$ be a quadratic $k$-algebra over $\mathcal{Q}$. Conjecture 5.2 holds if $n\leqslant 3$.

Proof 5.3**.**

Assume that $A$ is $n$-Calabi-Yau and $n$-Kc-Calabi-Yau. We can assume that $n\geqslant 2$. Since $\mathop{\rm HK}\nolimits^{p}(A,A^{e})\cong\mathop{\rm HH}\nolimits^{p}(A,A^{e})=0$ when $p=0$ and $p=1$, we have $H_{n}(K(A))\cong H_{n-1}(K(A))=0$, hence $A$ is Koszul if $n=2$. When $n=3$, we also have $H_{1}(K(A))=0$ because the complex $K(A)$ is always exact in degree $1$, therefore $A$ is also Koszul in this case.

5.3 Strong Kc-Calabi-Yau algebras

{Df}

Let $A$ be an $n$-Kc-Calabi-Yau algebra over $\mathcal{Q}$. The image $c\in\mathop{\rm HK}\nolimits_{n}(A)$ of the unit $1$ of the algebra $A$ by the isomorphism $\mathop{\rm HK}\nolimits^{0}(A)\cong\mathop{\rm HK}\nolimits_{n}(A)$ in Theorem 5.1 is called the fundamental class of the $n$-Kc-Calabi-Yau algebra $A$.

We shall now define strong Kc-Calabi-Yau algebras. For this, we need the DG algebra $\tilde{A}=\mathop{\rm Hom}\nolimits_{A^{e}}(K(A),A))$ and we work with DG $\tilde{A}$-bimodules in $A\text{-{Bimod}}$, as defined in Section 3 and recalled below. The point is that $K(A)$ is such a DG $\tilde{A}$-bimodule in $A\text{-{Bimod}}$ (Proposition 3.3). Denote by $\mathcal{C}(\tilde{A},A\text{-{Bimod}})$ and \mathcal{C}(\tilde{A},\text{{Vect}{}_{\mathbb{F}}}) the category of DG $\tilde{A}$-bimodules in $A\text{-{Bimod}}$ and VectF respectively [45]. Remark that a DG $\tilde{A}$-bimodule in VectF is just a DG $\tilde{A}$-bimodule.

{Df}

A DG $\tilde{A}$-bimodule $C$ in $A\text{-{Bimod}}$ is a chain complex in $A\text{-{Bimod}}$ (as usual, $C$ can be viewed as a cochain complex) endowed with a DG $\tilde{A}$-bimodule structure such that the bimodule actions of $A$ and $\tilde{A}$ are compatible.

For any $A$-bimodule $M$, $\mathop{\rm Hom}\nolimits_{A^{e}}(C,M)$ is a (cochain) DG $\tilde{A}$-bimodule in VectF (in $A\text{-{Bimod}}$ when $M=A^{e}$) for the following actions

[TABLE]

where $f:A\otimes_{k}W_{p}\otimes_{k}A\rightarrow A$, $u:C_{q}\rightarrow M$ and $x\in C_{p+q}$. Note that $x.f$ and $f.x$ are in $C_{q}$ by the graded actions of $\tilde{A}$ on $C$. If $C=K(A)$, we recover the cup actions, that is, $f.u=f\underset{K}{\smile}u$ and $u.f=u\underset{K}{\smile}f$. In particular, $\mathop{\rm Hom}\nolimits_{A^{e}}(K(A),A^{e})$ is a DG $\tilde{A}$-bimodule in $A\text{-{Bimod}}$.

Similarly, for any cochain DG $\tilde{A}$-bimodule $C^{\prime}$ in $A\text{-{Bimod}}$, $M\otimes_{A^{e}}C^{\prime}$ is a cochain DG $\tilde{A}$-bimodule in VectF for the following actions

[TABLE]

where $f\in\tilde{A}$, $m\in M$ and $u\in C^{\prime}$.

The bounded derived categories $\mathcal{D}^{b}(\tilde{A},A\text{-{Bimod}})$ and \mathcal{D}^{b}(\tilde{A},\text{{Vect}{}{\mathbb{F}}}) are defined in [45, Definition 7.2.7, Definition 7.3.3]. Unfortunately, it is not clear to us if the functors \mathop{\rm Hom}\nolimits_{A^{e}}(-,M):\mathcal{C}^{b}(\tilde{A},A\text{-{Bimod}})\rightarrow\mathcal{C}^{b}(\tilde{A},\text{{Vect}{}{\mathbb{F}}}) and M\otimes_{A^{e}}-:\mathcal{C}^{b}(\tilde{A},A\text{-{Bimod}})\rightarrow\mathcal{C}^{b}(\tilde{A},\text{{Vect}{}_{\mathbb{F}}}) can be derived. Note that the first one takes values in $\mathcal{C}^{b}(\tilde{A},A\text{-{Bimod}})$ when $M=A^{e}$.

{Df}

Let $A$ be a Kc-Calabi-Yau algebra of dimension $n$. Then $A$ is said to be strong $n$-Kc-Calabi-Yau if the derived functor of the endofunctor $\mathop{\rm Hom}\nolimits_{A^{e}}(-,A^{e})$ of $\mathcal{C}^{b}(\tilde{A},A\text{-{Bimod}})$ exists and if $\mathop{\rm RHom}\nolimits_{A^{e}}(K(A),A^{e})\cong K(A)[-n]$ in $\mathcal{D}^{b}(\tilde{A},A\text{-{Bimod}})$.

The preprojective algebras of connected graphs distinct from $\mathrm{A}_{1}$ and $\mathrm{A}_{2}$ are strong $2$-Kc-Calabi-Yau algebras if they satisfy the first property in this definition. In fact, using Proposition 4.4, $\theta_{A^{e}}$ provides then an isomorphism $\mathop{\rm RHom}\nolimits_{A^{e}}(K(A),A^{e})\rightarrow K(A)[-2]$ in $\mathcal{D}^{b}(\tilde{A},A\text{-{Bimod}})$.

**Theorem **\theDf

Let $A$ be a Kc-Calabi-Yau algebra of dimension $n$ over $\mathcal{Q}$, with fundamental class $c$. We assume that $A$ is strong Kc-Calabi-Yau and that the derived functors of the functors $\mathop{\rm Hom}\nolimits_{A^{e}}(-,A)$ and $A\otimes_{A^{e}}-$ from $\mathcal{C}^{b}(\tilde{A},A\text{-{Bimod}})$ to \mathcal{C}^{b}(\tilde{A},\text{{Vect}{}_{\mathbb{F}}}) exist. Then

[TABLE]

is an isomorphism of $\mathop{\rm HK}\nolimits^{\bullet}(A)$-bimodules, inducing an isomorphism of $\mathop{\rm HK}\nolimits^{\bullet}_{hi}(A)$-bimodules from $\mathop{\rm HK}\nolimits^{\bullet}_{hi}(A)$ to $\mathop{\rm HK}\nolimits^{hi}_{n-\bullet}(A)$. For all $\alpha\in\mathop{\rm HK}\nolimits^{p}(A)$, we have $c\underset{K}{\frown}\alpha=(-1)^{np}\alpha\underset{K}{\frown}c$.

Proof 5.4**.**

Following the proof of Theorem 5.1, we are interested in the morphism of cochain complexes

[TABLE]

when the bounded chain complex $C$ of $A$-bimodules is moreover a DG $\tilde{A}$-bimodule in $A\text{-{Bimod}}$. We prove now that $\phi_{M}$ is a morphism of DG $\tilde{A}$-bimodules in VectF, that is, a morphism in the category \mathcal{C}^{b}(\tilde{A},\text{{Vect}{}_{\mathbb{F}}}) whose objects are viewed as cochain complexes. For this, we need only prove that $\phi_{M}$ is a morphism of $\tilde{A}$-bimodules. Let us prove that $\phi_{M}$ is left $\tilde{A}$-linear, the right linearity being similar. For $f:A\otimes_{k}W_{p}\otimes_{k}A\rightarrow A$, $u:C_{q}\rightarrow A^{e}$ and $x\in C_{p+q}$, we have

[TABLE]

while $f.(\phi_{M}(m\otimes_{A^{e}}u))(x)=(-1)^{p}\phi_{M}(m\otimes_{A^{e}}u)(x.f)=(-1)^{p}m.(u(x.f))$, which is what we want.

Continuing as in the proof of Theorem 5.1, the functors $F$, $G$ and $H$ induce functors on the complexes with enriched structures. Precisely, $F$ and $G$ are now functors from $\mathcal{C}^{b}(\tilde{A},A\text{-{Bimod}})$ to \mathcal{C}^{b}(\tilde{A},\text{{Vect}{}_{\mathbb{F}}}), and $H$ is now an endofunctor of $\mathcal{C}^{b}(\tilde{A},A\text{-{Bimod}})$. Under these notations, $\phi_{M}$ defines a natural transformation $\phi_{M}:H\circ G\Rightarrow F$.

We specialise to $M=A$. The assumptions in the theorem show that the derived functors of $F$, $G$ and $H$ exist, so that we can derive the natural transformation $\phi_{A}$ [45]. Then for any bounded chain complex DG $\tilde{A}$-bimodule $C$ in $A\text{-{Bimod}}$ formed by finitely generated projective $A$-bimodules, we obtain an isomorphism

[TABLE]

in \mathcal{D}^{b}(\tilde{A},\text{{Vect}{}_{\mathbb{F}}}). Applying this to $C=K(A)$ and using Definition 5.3, we get

[TABLE]

in \mathcal{D}^{b}(\tilde{A},\text{{Vect}{}_{\mathbb{F}}}). Taking homology, we deduce an isomorphism $\mathop{\rm HK}\nolimits^{\bullet}(A)\cong\mathop{\rm HK}\nolimits_{n-\bullet}(A)$ of graded bimodules over the graded algebra $H(\tilde{A})=\mathop{\rm HK}\nolimits^{\bullet}(A)$. Denote this isomorphism by $\psi$.

The fact that $\psi$ is a morphism of graded $\mathop{\rm HK}\nolimits^{\bullet}(A)$-bimodules translates as

[TABLE]

for any $\alpha\in\mathop{\rm HK}\nolimits^{p}(A)$ and $\beta\in\mathop{\rm HK}\nolimits^{\bullet}(A)$. In accordance with Definition 5.3, define $c\in\mathop{\rm HK}\nolimits_{n}(A)$ by $c=\psi(1)$ where $1\in\mathop{\rm HK}\nolimits^{0}(A)$ is the unit of $A$. Applying identities (3) to the trivial equalities $\alpha=1\underset{K}{\smile}\alpha=\alpha\underset{K}{\smile}1$, we obtain

[TABLE]

Finally $\psi$ is a morphism of complexes for higher (co)homologies since we have

[TABLE]

Then $H(\psi):\mathop{\rm HK}\nolimits^{\bullet}_{hi}(A)\rightarrow\mathop{\rm HK}\nolimits^{hi}_{n-\bullet}(A)$ is an isomorphism of $\mathop{\rm HK}\nolimits^{\bullet}_{hi}(A)$-bimodules.

Except in some particular cases, $\overline{\mathfrak{e}}_{A}\in\mathop{\rm HK}\nolimits^{1}(A)$ is non-zero, so that

[TABLE]

is non-zero in $\mathop{\rm HK}\nolimits_{n-1}(A)$. Therefore $c\in\mathop{\rm HK}\nolimits_{n}(A)$ is not a cycle for higher Koszul homology and the isomorphism $H(\psi)$ cannot be naturally expressed as a cap action. As suggested by the preprojective algebras (Subsection 4.3), the class $\psi(\overline{\mathfrak{e}}_{A})$ should be of interest for further investigations.

It is also interesting to remark that the identities (3) involving the isomorphism $\psi$ imply that the graded algebra $\mathop{\rm HK}\nolimits^{\bullet}(A)$ is commutative if and only if the graded $\mathop{\rm HK}\nolimits^{\bullet}(A)$-bimodule $\mathop{\rm HK}\nolimits_{\bullet}(A)$ is symmetric. As seen in Corollary 4.2, we have a stronger result for the preprojective algebras.

6 Koszul calculus of the preprojective algebras of Dynkin ADE type

We shall determine in this section the Koszul calculus and the higher Koszul calculus of any non-Koszul preprojective algebra $A$, that is, an algebra of type A, D or E with at least $3$ vertices.

We first give some general facts and notation.

1. (N1)

We shall use the dimensions of the Hochschild cohomology and homology spaces of $A$ which can be obtained in all characteristics as a consequence of the work of Etingof, Eu and Schedler in [24, Theorem 3.2.7] and [21]. In particular, by [24, Lemma 3.2.17] the centre of $A$ is independent of the characteristic of $\mathbb{F}$. Bases of the Hochschild (co)homology spaces in characteristic zero induce free subsets of the Hochschild (co)homology spaces in positive characteristic, but there may be some extra basis elements in some cases. 2. (N2)

We know from Corollary 4.2 that the cup product on the Koszul cohomology of a preprojective algebra is graded commutative and that the cap product is graded symmetric. Moreover, it follows from Theorem 4.3 that the cap product can be obtained from the cup product. Indeed, if $f\in\mathop{\rm HK}\nolimits^{p}(A)$ and $x\in\mathop{\rm HK}\nolimits_{q}(A)$, we have

[TABLE] 3. (N3)

Let $X$ and $Y$ be $\mathbb{N}$-graded spaces and let $f:X\rightarrow Y$ be a homogeneous map of degree $1$. Let $y_{1},\ldots,y_{p}$ be elements of pairwise different degrees. Then if $\sum_{i=1}^{p}y_{i}\in\mathop{\rm Im}\nolimits f$, at least one of the $y_{i}$ is in $\mathop{\rm Im}\nolimits f$. We shall use this in the following context. The differentials $b_{K}^{1}$ and $b_{K}^{2}$ are homogeneous of weight $1$. If we have a set of cocycles of pairwise different coefficient weights, that are not coboundaries, then they are linearly independent up to coboundaries, that is, they represent linearly independent cohomology classes. This also applies if some of the elements have the same weight but we already know that these elements are linearly independent up to coboundaries. 4. (N4)

We shall use the map $\kappa\colon A\rightarrow A$ constructed as follows. Let $A$ be a preprojective algebra over a graph $\Delta$; let $\mkern 2.0mu\overline{\mkern-2.0muQ\mkern 1.0mu}\mkern-1.0mu$ be its quiver. Consider the map $\mkern 2.0mu\overline{\mkern-2.0muQ\mkern 1.0mu}\mkern-1.0mu_{1}\rightarrow\mkern 2.0mu\overline{\mkern-2.0muQ\mkern 1.0mu}\mkern-1.0mu_{1}$ that sends $a$ to $a^{*}$. It induces an anti-automorphism $\kappa$ of $A$ such that $\kappa(e_{i})=e_{i}$ for all $i\in Q_{0}$ (since $\kappa$ sends the relation $\sigma_{i}=\sum_{\begin{subarray}{c}a\in\mkern 2.0mu\overline{\mkern-2.0muQ\mkern 1.0mu}\mkern-1.0mu_{1}\\ \mathfrak{t}(a)=i\end{subarray}}\varepsilon(a)aa^{*}$ to itself). 5. (N5)

We shall be using the Nakayama automorphism $\nu$ of $A$ defined by Brenner, Butler and King for all preprojective algebras of Dynkin ADE type [10, Section 4].

In order to describe it, we need the Nakayama permutation $\mkern 2.0mu\overline{\mkern-2.0mu\nu\mkern 1.0mu}\mkern-1.0mu$ on the set of vertices of $\Delta$. It is known that $\mkern 2.0mu\overline{\mkern-2.0mu\nu\mkern 1.0mu}\mkern-1.0mu=\mathop{\rm id}\nolimits$ if $\Delta$ is $\mathrm{D}_{n}$ with $n$ even or if $\Delta$ is $\mathrm{E}_{7}$ or $\mathrm{E}_{8}$, and that otherwise $\mkern 2.0mu\overline{\mkern-2.0mu\nu\mkern 1.0mu}\mkern-1.0mu$ is induced by the unique graph automorphism of order $2$.

If $\alpha$ is an arrow in $\mkern 2.0mu\overline{\mkern-2.0muQ\mkern 1.0mu}\mkern-1.0mu_{1}$, let $\beta$ be the unique arrow from $\mkern 2.0mu\overline{\mkern-2.0mu\nu\mkern 1.0mu}\mkern-1.0mu(\mathfrak{s}(\alpha))$ to $\mkern 2.0mu\overline{\mkern-2.0mu\nu\mkern 1.0mu}\mkern-1.0mu(\mathfrak{t}(\alpha))$. The Nakayama automorphism of [10] is described as follows:

[TABLE]

Given a basis of a selfinjective quiver algebra $A=\mathbb{F}Q/I$ consisting of paths and containing a basis $\left\{\pi_{i}\,;\,i\in Q_{0}\right\}$ of the socle of $A$, an explicit construction of an associative non-degenerate bilinear form on $A$ was given in [46, Proposition 3.15] (see also Subsection 6.6). We have chosen in each case such a basis so that $\nu$ is the Nakayama automorphism corresponding to this bilinear form (characterised on the arrows $\alpha$ in $\mkern 2.0mu\overline{\mkern-2.0muQ\mkern 1.0mu}\mkern-1.0mu_{1}$ by $y\alpha=\pi_{\mathfrak{s}(\alpha)}\Leftrightarrow\nu(\alpha)y=\pi_{\mathfrak{t}(\alpha)}$ for all the basis elements $y$). 6. (N6)

When we define a cochain $f\in\mathop{\rm Hom}\nolimits_{k^{e}}(X,A)$ with $X\in\left\{k,V,A\right\}$, it will be implicit that if $f(x)$ is not defined for some $x\in X$ then $f(x)=0$. 7. (N7)

For any cochain $f\in\mathop{\rm Hom}\nolimits_{k^{e}}(W_{p},A)$, we shall set $\check{f}=\theta_{A}(f)\in A\otimes_{k^{e}}W_{2-p}$. 8. (N8)

Finally, given a Dynkin graph $\Delta$ and a ring $L$, we shall denote by $\Lambda_{L}$ the preprojective algebra of $\Delta$ over $L$, so that $A=\Lambda_{\mathbb{F}}$.

6.1 Koszul calculus for preprojective algebras of type A

The preprojective algebra $A$ of type $A_{n}$ is defined by the quiver

[TABLE]

subject to the relations

[TABLE]

The Nakayama automorphism $\nu$ of $A$ described in 5 is given by $\nu(e_{i})=e_{n-1-i}$, $\nu(a_{i})=a_{n-2-i}^{*}$ and $\nu(a_{i}^{*})=a_{n-2-i}$.

Erdmann and Snashall have given in [18] a basis $B$ of $A$. We shall only need the sets $e_{i}Be_{i}$, $e_{i}Be_{i+1}$ and $e_{i+1}Be_{i}$, which can be rewritten as follows: set ${m_{A}}=\left\lfloor{\frac{n-1}{2}}\right\rfloor$; then

[TABLE]

For each $i$, $Ae_{i}$ contains precisely one basis element of maximal length $n-1$, which is

[TABLE]

They form a basis of the socle of $A$.

6.1.1 The Koszul cohomology and homology spaces in type A

The spaces $\mathop{\rm HK}\nolimits^{0}(A)=\mathop{\rm HH}\nolimits^{0}(A)=Z(A)$ and $\mathop{\rm HK}\nolimits^{1}(A)=\mathop{\rm HH}\nolimits^{1}(A)$ are known from [18]. Therefore we only need to compute $\mathop{\rm HK}\nolimits^{2}(A)$. Recall our assumption that $n\geqslant 3$; then by Theorem 4.2 all the elements in $\mathop{\rm Hom}\nolimits_{k^{e}}(R,A)$ are cocycles. Moreover, using Theorem 4.3 and [24, Theorem 3.2.7], we have $\dim\mathop{\rm HK}\nolimits^{2}(A)=\dim\mathop{\rm HK}\nolimits_{0}(A)=\dim\mathop{\rm HH}\nolimits_{0}(A)=n$. Since every element in $\mathop{\rm Im}\nolimits b_{K}^{2}$ has coefficient weight at least $1$, the $n$ cocycles $h_{i}$ defined by $h_{i}(\sigma_{j})=\delta_{ij}e_{i}$ for all $j$ are linearly independent modulo $\mathop{\rm Im}\nolimits b_{K}^{2}$. It follows that they form a basis of $\mathop{\rm HK}\nolimits^{2}(A)$.

Combining with the results from [18], we have the following result.

**Proposition **\theDf

Let $A$ be a preprojective algebra of type An.

A basis of $\mathop{\rm HK}\nolimits^{0}(A)$ is given by the set $\left\{z_{\ell};0\leqslant\ell\leqslant{m_{A}}\right\}$ with $z_{0}=1$ and $z_{\ell}=\sum_{i=1}^{n-2}(a_{i}^{*}a_{i})^{\ell}=\sum_{i=\ell}^{n-1-\ell}(a_{i}^{*}a_{i})^{\ell}=z_{1}^{\ell}$ for $1\leqslant\ell\leqslant{m_{A}}.$

A basis of $\mathop{\rm HK}\nolimits^{1}(A)$ is given by the set $\left\{\mkern 2.0mu\overline{\mkern-2.0mu\zeta\mkern 1.0mu}\mkern-1.0mu_{\ell};0\leqslant\ell\leqslant n-2-{m_{A}}\right\}$, where $\zeta_{\ell}\in\mathop{\rm Hom}\nolimits_{k^{e}}(V,A)$ is defined by $\zeta_{\ell}(a_{i})=a_{i}(a_{i}^{*}a_{i})^{\ell}$ for all $i$ (or for $\ell\leqslant i\leqslant n-2-\ell$).

A basis of $\mathop{\rm HK}\nolimits^{2}(A)$ is given by the set $\left\{\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{i};0\leqslant i\leqslant n-1\right\}$ where $h_{i}\in\mathop{\rm Hom}\nolimits_{k^{e}}(R,A)$ defined by $h_{i}(\sigma_{j})=\delta_{ij}e_{i}$ for all $j$.

As a consequence of Theorem 4.3, we obtain bases of the Koszul homology spaces.

**Corollary **\theDf

A basis of $\mathop{\rm HK}\nolimits_{0}(A)$ is given by the set $\left\{\mkern 2.0mu\overline{\mkern-2.0mu\check{h}\mkern 1.0mu}\mkern-1.0mu_{i};0\leqslant i\leqslant n-1\right\}$ where $\check{h}_{i}=e_{i}\otimes e_{i}$.

A basis of $\mathop{\rm HK}\nolimits_{1}(A)$ is given by the set $\left\{\mkern 2.0mu\overline{\mkern-2.0mu\check{\zeta}\mkern 1.0mu}\mkern-1.0mu_{\ell};0\leqslant\ell\leqslant n-{m_{A}}-2\right\}$ where $\check{\zeta}_{\ell}=\sum_{i=0}^{n-2}a_{i}(a_{i}^{*}a_{i})^{\ell}\otimes a_{i}^{*}=\sum_{i=\ell}^{n-2-\ell}a_{i}(a_{i}^{*}a_{i})^{\ell}\otimes a_{i}^{*}$. A basis of $\mathop{\rm HK}\nolimits_{2}(A)$ is given by the set $\left\{\check{z}_{\ell};0\leqslant\ell\leqslant{m_{A}}\right\}$ where $\check{z}_{\ell}=\sum_{i=0}^{n-1}(a_{i}^{*}a_{i})^{\ell}\otimes\sigma_{i}$.

Note that $\check{z}_{0}=\omega_{0}$ is the fundamental class.

6.1.2 Cup and cap products

We know from Corollary 4.2 that the cup product on $\mathop{\rm HK}\nolimits^{\bullet}(A)$ is graded-commutative. The following result gives all the non zero cup products of elements in $\mathop{\rm HK}\nolimits^{\bullet}(A)$.

**Proposition **\theDf

Let $A$ be a preprojective algebra of type An. Up to graded commutativity, the non zero cup products in $\mathop{\rm HK}\nolimits^{\bullet}(A)$ are given by

[TABLE]

Proof 6.1**.**

The first cup product is clear and the other cup products in the statement only involve in $\mathop{\rm HH}\nolimits^{0}(A)$ and $\mathop{\rm HH}\nolimits^{1}(A)$, therefore they are known from [18].

The basis elements of $\mathop{\rm HK}\nolimits^{2}(A)$ have coefficient weight [math], and $b_{K}^{2}$ is homogeneous of weight 1, therefore any element that has positive coefficient weight must be a coboundary. The other cup products (that all vanish) follow from this.

We now deduce the cap products from (5).

**Corollary **\theDf

Up to graded symmetry, the non zero cap products are the following.

[TABLE]

6.1.3 Higher Koszul cohomology and homology

We start with a lemma giving the cohomology class of the fundamental $1$-cocycle.

**Lemma **\theDf

The cohomology class of $\mathfrak{e}_{A}$ is equal to the cohomology class of $2\zeta_{0}.$

Proof 6.2**.**

Let $\zeta^{*}_{0}\in\mathop{\rm Hom}\nolimits_{k^{e}}(V,A)$ be the cocycle defined by $\zeta^{*}_{0}(a_{i}^{*})=a_{i}^{*}$ for all $i\in Q_{0}$. Since $\mathfrak{e}_{A}=\zeta_{0}+\zeta_{0}^{*}$, we must prove that $\zeta_{0}^{*}-\zeta_{0}$ is a coboundary.

Consider $v=\sum_{i=0}^{n-2}\sum_{j=i}^{n-2}e_{i}\in\bigoplus_{i\in Q_{0}}e_{i}Ae_{i}\cong\mathop{\rm Hom}\nolimits_{k^{e}}(k,A)$. Then $b_{K}^{1}(v)=\zeta_{0}^{*}-\zeta_{0}$, as required.

As a consequence, the complex defining the higher Koszul cohomology is

[TABLE]

with $\partial_{\smile}^{1}(z_{\ell})=2\mkern 2.0mu\overline{\mkern-2.0mu\zeta\mkern 1.0mu}\mkern-1.0mu_{\ell}$ for $0\leqslant\ell\leqslant n-{m_{A}}-2$ and $\partial_{\smile}^{2}=0.$ We then have the following higher Koszul cohomology.

**Proposition **\theDf

Let $A$ be a preprojective algebra of type An. If $\mathop{\rm char}\nolimits(\mathbb{F})=2$, then $\mathop{\rm HK}\nolimits^{\bullet}_{hi}(A)=\mathop{\rm HK}\nolimits^{\bullet}(A).$

If $\mathop{\rm char}\nolimits(\mathbb{F})\neq 2$ and $n$ is even, then

[TABLE]

Finally, if $\mathop{\rm char}\nolimits(\mathbb{F})\neq 2$ and $n$ is odd, then

[TABLE]

Higher Koszul homology can then be deduced using duality (Theorem 4.3).

**Corollary **\theDf

If $\mathop{\rm char}\nolimits(\mathbb{F})=2$, then $\mathop{\rm HK}\nolimits_{\bullet}^{hi}(A)=\mathop{\rm HK}\nolimits_{\bullet}(A).$

If $\mathop{\rm char}\nolimits(\mathbb{F})\neq 2$ and $n$ is even, then

[TABLE]

Finally, if $\mathop{\rm char}\nolimits(\mathbb{F})\neq 2$ and $n$ is odd, then

[TABLE]

6.2 Koszul calculus for preprojective algebras of type D

The preprojective algebra $A$ of type $D_{n}$ is defined by the quiver

[TABLE]

subject to the relations

[TABLE]

The Nakayama automorphism $\nu$ of $A$ described in 5 is given by $\nu(e_{i})=e_{i}$, $\nu(a_{i})=-a_{i}^{*}$ and $\nu(a_{i}^{*})=a_{i}$ if $n$ is even of if $n$ is odd and $i\geqslant 2$, and it exchangess $e_{0}$ and $e_{1}$, $a_{0}$ and $-a_{1}$, and $a_{0}^{*}$ and $a_{1}^{*}$ if $n$ is odd.

Eu has given in [22] a basis $B$ of $A$. Set ${m_{D}}=\left\lfloor{\frac{n-2}{2}}\right\rfloor$ and $u=n-{m_{D}}-2.$ We shall only need bases of the $e_{j}Ae_{i}$ when $i$ and $j$ are equal or adjacent vertices, which can be rewritten as follows:

[TABLE]

For each $i$, $Ae_{i}$ contains precisely one basis element of maximal length $2(n-2)$, which is

[TABLE]

They form a basis of the socle of $A$.

6.2.1 The Koszul cohomology and homology spaces in type D

The centre of $A$ does not depend on the characteristic of $\mathbb{F}$ by fact 1 and was computed in [22] in characteristic [math]. A basis of $\mathop{\rm HH}\nolimits^{1}(\Lambda_{\mathbb{C}})$ was determined in [22]. Moreover, $\dim\mathop{\rm HK}\nolimits^{1}(A)=\dim\mathop{\rm HH}\nolimits^{1}(A)$, which is equal to $\dim\mathop{\rm HH}\nolimits^{1}(\Lambda_{\mathbb{C}})=n$ if $\mathop{\rm char}\nolimits(\mathbb{F})\neq 2$ and to $\dim\mathop{\rm HH}\nolimits^{1}(\Lambda_{\mathbb{C}})+m$ if $\mathop{\rm char}\nolimits(\mathbb{F})=2$ by [24].

It also follows from Theorem 4.3 and [24] that $\dim\mathop{\rm HK}\nolimits^{2}(A)=\dim\mathop{\rm HK}\nolimits_{0}(A)=\dim\mathop{\rm HH}\nolimits_{0}(A)$, which is equal to $n-{m_{D}}-2$ if $\mathop{\rm char}\nolimits(\mathbb{F})\neq 2$ and to $n-2$ if $\mathop{\rm char}\nolimits(\mathbb{F})=2$.

In order to give bases of the $\mathop{\rm HK}\nolimits^{p}(A)$ for $p=0,1,2$, we define the following cochains:

• •

the elements $z_{0}=1$ and $z_{\ell}=(a_{0}^{*}a_{1}a_{1}^{*}a_{0})^{\ell}+(a_{1}^{*}a_{0}a_{0}^{*}a_{1})^{\ell}+\sum_{i=2}^{n-2}(a_{i}^{*}a_{i})^{2\ell}=z_{1}^{\ell}$ for $\ell>0$ in $A$. Note that if $n$ is even, then $z_{1}^{u}=z_{1}^{{m_{D}}}=-\pi_{0}+\pi_{1}$, but if $n$ is odd then $z_{1}^{u}=0$;

• •

the elements $\zeta_{\ell}\in\mathop{\rm Hom}\nolimits_{k^{e}}(V,A)$ with $0\leqslant\ell\leqslant u-1$ defined by $\zeta_{\ell}(a_{i})=a_{i}z_{\ell}$ for all $i$;

• •

the elements $\rho_{\ell}\in\mathop{\rm Hom}\nolimits_{k^{e}}(V,A)$ with $0\leqslant\ell\leqslant{m_{D}}-1$ where $\rho_{\ell}(a_{i})=(a_{2}^{*}a_{2})^{2\ell+1}a_{i}$ for $i=0,1$ and $\rho_{\ell}(a_{i}^{*})=a_{i}^{*}(a_{2}^{*}a_{2})^{2\ell+1}$ for $i=0,1$;

• •

the elements $h_{j}\in\mathop{\rm Hom}\nolimits_{k^{e}}(R,A)$ for $0\leqslant j\leqslant n-1$, where $h_{j}(\sigma_{i})=\delta_{ij}e_{j}$;

• •

the elements $\gamma_{\ell}\in\mathop{\rm Hom}\nolimits_{k^{e}}(R,A)$ for $1\leqslant\ell\leqslant{m_{D}}$ where $\gamma_{\ell}(\sigma_{0})=(a_{0}^{*}a_{1}a_{1}^{*}a_{0})^{\ell}$.

**Proposition **\theDf

Let $A$ be a preprojective algebra of type $D_{n}$.

1. \edefitit(i)

The elements in $\left\{\pi_{i};0\leqslant i\leqslant n-1\text{ and }\nu(e_{i})=e_{i}\right\}\cup\left\{z_{\ell};0\leqslant\ell\leqslant u-1\right\}$ form a basis of $\mathop{\rm HK}\nolimits^{0}(A).$ 2. \edefitit(ii)

If $\mathop{\rm char}\nolimits(\mathbb{F})\neq 2$, the $\mkern 2.0mu\overline{\mkern-2.0mu\zeta\mkern 1.0mu}\mkern-1.0mu_{\ell}$, for $0\leqslant\ell\leqslant u-1$, form a basis of $\mathop{\rm HK}\nolimits^{1}(A)$.

If $\mathop{\rm char}\nolimits(\mathbb{F})=2$, the $\mkern 2.0mu\overline{\mkern-2.0mu\zeta\mkern 1.0mu}\mkern-1.0mu_{\ell}$ for $0\leqslant\ell\leqslant u-1$ and the $\mkern 2.0mu\overline{\mkern-2.0mu\rho\mkern 1.0mu}\mkern-1.0mu_{\ell}$ for $0\leqslant\ell\leqslant{m_{D}}-1$ form a basis of $\mathop{\rm HK}\nolimits^{1}(A)$. 3. \edefitit(iii)

If $\mathop{\rm char}\nolimits(\mathbb{F})\neq 2$, the $\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{j}$ for $0\leqslant j\leqslant n-1$ form a basis of $\mathop{\rm HK}\nolimits^{2}(A)$.

If $\mathop{\rm char}\nolimits(\mathbb{F})=2$, the $\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{j}$ for $0\leqslant j\leqslant n-1$ and the $\mkern 2.0mu\overline{\mkern-2.0mu\gamma\mkern 1.0mu}\mkern-1.0mu_{\ell}$ for $1\leqslant\ell\leqslant{m_{D}}$ form a basis of $\mathop{\rm HK}\nolimits^{2}(A)$.

Proof 6.3**.**

The results for $\mathop{\rm HK}\nolimits^{0}(A)$ and, when $\mathop{\rm char}\nolimits(\mathbb{F})\neq 2$, for $\mathop{\rm HK}\nolimits^{1}(A)$ follow from the comments before the proposition.

Assume that $\mathop{\rm char}\nolimits(\mathbb{F})=2$. In order to prove the result for $\mathop{\rm HK}\nolimits^{1}(A)$, we must prove that the elements we have considered in $\mathop{\rm Hom}\nolimits_{k^{e}}(V,A)$ are cocycles that are linearly independent modulo coboundaries. It is in fact enough to prove that the $\rho_{\ell}$ are cocycles that are not coboundaries by fact 3.

First note that, at the level of cochains, $\rho_{\ell}=\rho_{0}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}z_{\ell}$. Therefore, to prove that $\rho_{\ell}$ is a cocycle, it is enough to prove that $\rho_{0}$ is a cocycle, and this is easy to check.

Since $\rho_{\ell}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}z_{{m_{D}}-1-\ell}=\rho_{{m_{D}}-1}$, in order to prove that $\rho_{\ell}$ is not a coboundary for all $\ell$, it is enough to prove that $\rho_{{m_{D}}-1}$ is not a coboundary. The map $\rho_{{m_{D}}-1}$ has coefficient weight $4{m_{D}}-1$. If $\rho_{{m_{D}}-1}$ is a coboundary, then it is the image of a morphism in $\mathop{\rm Hom}\nolimits_{k^{e}}(k,A)\cong\bigoplus_{i\in Q_{0}}e_{i}Ae_{i}$ whose coefficients are linear combinations of cycles in $A$ of weight $4{m_{D}}-2$, which are known. It is then straightforward to show that the image of any such morphism under $b_{K}^{1}$ is not equal to $\rho_{{m_{D}}-1}$.

For (iii), we first observe that every cochain in $\mathop{\rm Hom}\nolimits_{k^{e}}(R,A)$ is a cocycle. Moreover, the $h_{j}$ are $n$ cocycles that are clearly linearly independent modulo coboundaries (all coboundaries have coefficient weight at least equal to $1$). Therefore if $\mathop{\rm char}\nolimits(\mathbb{F})\neq 2$, the result follows.

If $\mathop{\rm char}\nolimits(\mathbb{F})=2$, it is enough to prove that the $\gamma_{\ell}$ are not coboundaries by fact 3. At the level of cochains, $\gamma_{\ell}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}z_{{m_{D}}-\ell}=\gamma_{m_{D}}$ for $1\leqslant\ell\leqslant{m_{D}}$, therefore it is enough to prove that $\gamma_{m_{D}}$ is not a coboundary. The map $\gamma_{m_{D}}$ has coefficient weight $4{m_{D}}$, therefore if $\gamma_{m_{D}}$ is a coboundary, then it is the image of a morphism in $\mathop{\rm Hom}\nolimits_{k^{e}}(V,A)$ whose coefficients are linear combinations of elements of weight $4{m_{D}}-2$ in $A$ between two adjacent vertices in $Q_{0}$, which are known. Here again, checking that that the image of any such morphism under $b_{K}^{2}$ is not equal to $\gamma_{m_{D}}$ is straightforward.

Koszul homology follows using duality (Theorem 4.3), as in Corollary 6.1.1.

6.2.2 Cup and cap products

We now determine the cup products of the elements in the bases of the Koszul cohomology spaces given above.

**Lemma **\theDf

For $1\leqslant i\leqslant n-2$, consider the cochains $u_{i},$ $v_{i}$ and $w_{i}$ in $\mathop{\rm Hom}\nolimits_{k^{e}}(R,A)$ defined by $u_{i}(\sigma_{j})=\delta_{ij}e_{i}z_{1}$, $v_{i}(\sigma_{j})=\delta_{ij}\pi_{i}$ and $w_{i}(\sigma_{j})=\delta_{ij}a_{i}^{*}a_{i}z_{1}$ for all $j\in Q_{0}$.

If $\mathop{\rm char}\nolimits(\mathbb{F})\neq 2$, the $u_{i}$, $v_{i}$ and $w_{i}$ are all coboundaries.

If $\mathop{\rm char}\nolimits(\mathbb{F})=2$, then the $u_{i}$ for $i\geqslant 2$ are coboundaries, and $\mkern 2.0mu\overline{\mkern-2.0muu\mkern 1.0mu}\mkern-1.0mu_{0}=\mkern 2.0mu\overline{\mkern-2.0muu\mkern 1.0mu}\mkern-1.0mu_{1}=\mkern 2.0mu\overline{\mkern-2.0mu\gamma\mkern 1.0mu}\mkern-1.0mu_{1}$. Moreover, if $n$ is odd, all the $v_{i}$ are coboundaries and if $n$ is even, then $\mkern 2.0mu\overline{\mkern-2.0muv\mkern 1.0mu}\mkern-1.0mu_{i}=\mkern 2.0mu\overline{\mkern-2.0mu\gamma\mkern 1.0mu}\mkern-1.0mu_{{m_{D}}}$ for all $i$. Finally, the $w_{i}$ are all coboundaries.

Proof 6.4**.**

Every element in $\mathop{\rm Hom}\nolimits_{k^{e}}(R,A)$ is a cocycle. Moreover, the differential $b_{K}^{2}$ is homogeneous of degree $1$ with respect to the coefficient weight, and the coefficient weight of all the basis elements in $\mathop{\rm HK}\nolimits^{2}(A)$ is a multiple of $4$, and is [math] if $\mathop{\rm char}\nolimits(\mathbb{F})\neq 2.$

It follows that if $\mathop{\rm char}\nolimits(\mathbb{F})\neq 2$, all the $u_{i},$ $v_{i}$ and $w_{i}$ must be coboundaries, and if $\mathop{\rm char}\nolimits(\mathbb{F})=2$, the $w_{i}$ are coboundaries and so are the $v_{i}$ if $n$ is odd.

Assume that $\mathop{\rm char}\nolimits(\mathbb{F})=2$. We must now study the $u_{i}$, as well as the $v_{i}$ when $n$ is even.

Note that $u_{n-2}=0$ and $u_{0}=\gamma_{1}$. For $0\leqslant i\leqslant n-3$, define $p_{i}\in\mathop{\rm Hom}\nolimits_{k^{e}}(V,A)$ by $p_{i}(a_{0})=a_{1}a_{1}^{*}a_{0}$, $p_{i}(a_{1})=a_{0}a_{0}^{*}a_{1}$ and $p_{i}(a_{i})=a_{i}a_{i}^{*}a_{i}$ if $i\geqslant 2$. Then, for $2\leqslant i\leqslant n-3$, we have $u_{i}=b_{K}^{2}\left(\sum_{j=i}^{n-3}p_{j}\right)$. Moreover, $b_{K}^{2}\left(\sum_{j=0}^{n-3}p_{j}\right)=u_{0}+u_{1}.$ It follows that the cohomology classes of $u_{0}$ and $u_{1}$ are both equal to that of $\gamma_{1}$.

We now turn to the $v_{i}.$ Note that since $n$ is even and $\mathop{\rm char}\nolimits(\mathbb{F})=2$, the map $v_{0}$ is the map $\gamma_{m_{D}}$, which is not a coboundary.

Define $q_{i}\in\mathop{\rm Hom}\nolimits_{k^{e}}(V,A)$ by $q_{i}(a_{i})=(a_{i+1}^{*}a_{i+1})^{n-i-2}a_{i}\cdots a_{2}a_{1}a_{1}^{*}a_{2}^{*}\cdots a_{i-1}^{*}$ for $1\leqslant i\leqslant n-2$ and $q_{0}(a_{0})=(a_{2}^{*}a_{2})^{n-3}$. Then, for $2\leqslant i\leqslant n-2$, we have $v_{i}-v_{1}=b_{K}^{2}\left(\sum_{j=1}^{i-1}q_{j}\right)$, and $v_{1}-v_{0}=b_{K}^{2}(q_{0}-q_{1})$. Therefore $\mkern 2.0mu\overline{\mkern-2.0muv\mkern 1.0mu}\mkern-1.0mu_{i}=\mkern 2.0mu\overline{\mkern-2.0muv\mkern 1.0mu}\mkern-1.0mu_{0}=\mkern 2.0mu\overline{\mkern-2.0mu\gamma\mkern 1.0mu}\mkern-1.0mu_{m_{D}}$ for all $i$.

We now give all the non zero cup products.

**Proposition **\theDf

Let $A$ be a preprojective algebra of type Dn. Up to graded commutativity, the non zero cup products of elements in $\mathop{\rm HK}\nolimits^{\bullet}(A)$ are:

[TABLE]

Proof 6.5**.**

We use the notation in Lemma 6.2.2.

For $\ell\geqslant 1$, we have $z_{\ell}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}h_{i}=z_{\ell-1}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}z_{1}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}h_{i}=z_{\ell-1}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}u_{i}$ and the result follows from Lemma 6.2.2.

Next, $\pi_{i}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}h_{j}=\delta_{ij}v_{j}$ and again the result is a consequence of Lemma 6.2.2.

Now assume that $\mathop{\rm char}\nolimits(\mathbb{F})=2$, so that the $\mkern 2.0mu\overline{\mkern-2.0mu\rho\mkern 1.0mu}\mkern-1.0mu_{\ell}$ occur in the basis of $\mathop{\rm HK}\nolimits^{1}(A).$ At the level of cochains, we have $\rho_{\ell_{1}}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}\rho_{\ell_{2}}=w_{2}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}z_{\ell_{1}+\ell_{2}}$, which is a coboundary.

The map $\rho_{\ell_{1}}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}\zeta_{\ell_{2}}=u_{2}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}z_{\ell_{1}+\ell_{2}+1}$ is also a coboundary, as required.

The remaining cup products are easy to compute. Note that the cup product in $\mathop{\rm HK}\nolimits_{0}(A)\cong Z(A)$ is the ordinary product, and that the elements $\pi_{i}$ are in the socle of $A$, hence are annihilated by the radical of $A$.

The cap products follow using duality, as in Corollary 6.1.2.

6.2.3 Higher Koszul (co)homology

As in the case of a preprojective algebra of type $A$, the cohomology class of the fundamental $1$-cocycle is equal to $2\mkern 2.0mu\overline{\mkern-2.0mu\zeta\mkern 1.0mu}\mkern-1.0mu_{0}$ so that $\partial_{\smile}^{1}(z_{\ell})=2\mkern 2.0mu\overline{\mkern-2.0mu\zeta\mkern 1.0mu}\mkern-1.0mu_{\ell}$ for $0\leqslant\ell\leqslant u-1$, $\partial_{\smile}^{1}(\pi_{i})=0$ and $\partial_{\smile}^{2}=0.$ We then have the following higher Koszul cohomology.

**Proposition **\theDf

Let $A$ be a preprojective algebra of type Dn.

1. \edefitit(i)

If $\mathop{\rm char}\nolimits(\mathbb{F})=2$, then $\mathop{\rm HK}\nolimits^{\bullet}_{hi}(A)=\mathop{\rm HK}\nolimits^{\bullet}(A).$ 2. \edefitit(ii)

If $\mathop{\rm char}\nolimits(\mathbb{F})\neq 2$, then

[TABLE]

Higher Koszul homology follows from Theorem 4.3 as in Corollary 6.1.3.

6.3 Koszul calculus for preprojective algebras of type E6

The preprojective algebra $A$ of type $E_{6}$ is defined by the quiver

[TABLE]

subject to the relations

[TABLE]

The Nakayama automorphism of 5 is given by

[TABLE]

To simplify notation, we shall denote by $c_{0}=a_{0}a_{0}^{*}$, $c_{2}=a_{2}a_{2}^{*}$ and $c_{3}=a_{3}^{*}a_{3}$ the three $2$-cycles at the vertex $3$.

The socle is the part of weight $10$ of $A$, and the set $\left\{\pi_{i};i\in Q_{0}\right\}$ where $\pi_{0}=a_{0}^{*}c_{3}^{2}c_{0}c_{3}a_{0}$, $\pi_{1}=a_{4}a_{3}c_{0}c_{3}c_{0}a_{2}a_{1}$, $\pi_{2}=a_{2}^{*}(c_{3}c_{0})^{2}a_{3}^{*}$, $\pi_{3}=c_{3}(c_{0}c_{3})^{2}$, $\pi_{4}=\kappa(\pi_{2})$ and $\pi_{5}=\kappa(\pi_{1})$.

6.3.1 The Koszul cohomology and homology spaces in type E6

We shall follow the same method as in types A and D, using the results from [24] and Theorem 4.3 to determine the dimensions of the spaces, and using results from [22] for the parts that are the same as in characteristic [math].

We define the following elements

• •

in $A$: $z_{0}=1$, $z_{6}=a_{1}^{*}a_{2}^{*}a_{3}^{*}a_{3}a_{2}a_{1}+a_{2}^{*}c_{3}^{2}a_{2}-c_{0}c_{3}c_{0}+a_{3}c_{2}^{2}a_{3}^{*}+a_{4}a_{3}a_{2}a_{2}^{*}a_{3}^{*}a_{4}^{*}$ and $z_{8}=a_{2}^{*}c_{0}c_{3}c_{0}a_{2}+c_{0}c_{3}^{2}c_{0}+a_{3}c_{0}c_{3}c_{0}a_{3}^{*}$;

• •

in $\mathop{\rm Hom}\nolimits_{k^{e}}(V,A)$: the maps $\zeta_{\ell}$ defined by $\zeta_{\ell}(a_{i})=a_{i}z_{\ell}$ for $\ell\in\left\{0,6,8\right\}$, the map $\rho_{3}$ defined by $\rho_{3}(a_{2})=c_{0}a_{2}$, $\rho_{3}(a_{3})=a_{3}c_{3}$ and $\rho_{3}(a_{2}^{*})=a_{2}^{*}c_{3}$, and the map $\rho_{5}$ defined by $\rho_{5}(a_{0})=c_{2}c_{3}a_{0}$, $\rho_{5}(a_{1})=a_{2}^{*}c_{0}a_{2}a_{1}$, $\rho_{5}(a_{2})=c_{2}^{2}a_{2}$, $\rho_{5}(a_{0}^{*})=-a_{0}^{*}c_{3}^{2}$, $\rho_{5}(a_{1}^{*})=-a_{1}^{*}a_{2}^{*}c_{0}a_{2}$ and $\rho_{5}(a_{2}^{*})=a_{2}^{*}c_{3}^{2}$;

• •

in $\mathop{\rm Hom}\nolimits_{k^{e}}(R,A)$: the maps $h_{j}$ defined for $0\leqslant j\leqslant 5$ by $h_{j}(\sigma_{i})=\delta_{ij}e_{j}$ for all $i$, the map $\gamma_{4}$ defined by $\gamma_{4}(\sigma_{0})=a_{0}^{*}c_{3}a_{0}$ and the map $\gamma_{6}$ defined by $\gamma_{6}(\sigma_{0})=a_{0}^{*}c_{3}^{2}a_{0}$.

We shall use the following lemma.

**Lemma **\theDf

Assume that $\mathop{\rm char}\nolimits(\mathbb{F})=3$. Let $\gamma\in\mathop{\rm Hom}\nolimits_{k^{e}}(R,A)$ be an element of coefficient weight $6$, so that

[TABLE]

Then $\gamma$ is a coboundary if, and only if, $\sum_{i=0}^{5}\lambda_{i}+\sum_{i=2}^{4}\lambda_{i}^{\prime}+\lambda_{3}^{\prime\prime}=0$.

Proof 6.6**.**

The proof is straightforward, once we know that a cochain of weight $5$ takes its values in $A_{5}=E\oplus\kappa(E)$, where $E$ is the space spanned by $c_{3}^{2}a_{0}$, $c_{0}c_{3}a_{0}$, $a_{2}^{*}c_{0}a_{2}a_{1},c_{3}^{2}a_{2}$, $c_{0}c_{3}a_{2}$, $a_{3}c_{0}c_{3}$, $a_{3}c_{3}c_{0}$, $a_{4}a_{3}c_{0}a_{3}^{*}$.

**Proposition **\theDf

Let $A$ be a preprojective algebra of type E6.

1. \edefitit(i)

The elements in $\left\{z_{0},z_{6},z_{8},\pi_{0},\pi_{3}\right\}$ form a basis of $\mathop{\rm HK}\nolimits^{0}(A)$. 2. \edefitit(ii)

If $\mathop{\rm char}\nolimits(\mathbb{F})\not\in\left\{2,3\right\}$, the elements in $\left\{\mkern 2.0mu\overline{\mkern-2.0mu\zeta\mkern 1.0mu}\mkern-1.0mu_{\ell};\ell=0,6,8\right\}$ form a basis of $\mathop{\rm HK}\nolimits^{1}(A)$.

If $\mathop{\rm char}\nolimits(\mathbb{F})=2$, the elements in $\left\{\mkern 2.0mu\overline{\mkern-2.0mu\zeta\mkern 1.0mu}\mkern-1.0mu_{\ell};\ell=0,6,8\right\}\cup\left\{\mkern 2.0mu\overline{\mkern-2.0mu\rho\mkern 1.0mu}\mkern-1.0mu_{3}\right\}$ form a basis of $\mathop{\rm HK}\nolimits^{1}(A)$.

If $\mathop{\rm char}\nolimits(\mathbb{F})=3$, the elements in $\left\{\mkern 2.0mu\overline{\mkern-2.0mu\zeta\mkern 1.0mu}\mkern-1.0mu_{\ell};\ell=0,6,8\right\}\cup\left\{\mkern 2.0mu\overline{\mkern-2.0mu\rho\mkern 1.0mu}\mkern-1.0mu_{5}\right\}$ form a basis of $\mathop{\rm HK}\nolimits^{1}(A)$. 3. \edefitit(iii)

If $\mathop{\rm char}\nolimits(\mathbb{F})\not\in\left\{2,3\right\}$, the elements in $\left\{\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{j};j\in Q_{0}\right\}$ form a basis of $\mathop{\rm HK}\nolimits^{2}(A)$.

If $\mathop{\rm char}\nolimits(\mathbb{F})=2$, the elements in $\left\{\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{j};j\in Q_{0}\right\}\cup\left\{\mkern 2.0mu\overline{\mkern-2.0mu\gamma\mkern 1.0mu}\mkern-1.0mu_{4}\right\}$ form a basis of $\mathop{\rm HK}\nolimits^{2}(A)$.

If $\mathop{\rm char}\nolimits(\mathbb{F})=3$, the elements in $\left\{\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{j};j\in Q_{0}\right\}\cup\left\{\mkern 2.0mu\overline{\mkern-2.0mu\gamma\mkern 1.0mu}\mkern-1.0mu_{6}\right\}$ form a basis of $\mathop{\rm HK}\nolimits^{2}(A)$.

Proof 6.7**.**

The centre was given in [22], so we have (i).

For $\mathop{\rm HK}\nolimits^{1}(A)$ and $\mathop{\rm HK}\nolimits^{2}(A)$, the number of elements in the statement is equal to the dimension of the corresponding cohomology space. Moreover, all the elements in the statement are indeed cocycles.

If $\mathop{\rm char}\nolimits(\mathbb{F})$ is not $2$ or $3$, a basis of $\mathop{\rm HK}\nolimits^{1}(A)=\mathop{\rm HH}\nolimits^{1}(A)$ was given in [22]. It consists of the classes of the $\zeta^{\prime}_{\ell}$ with $\ell\in\left\{0,6,8\right\}$ where $\zeta^{\prime}_{\ell}(a_{i})=a_{i}z_{\ell}$ for $0\leqslant i\leqslant 2$ and $\zeta^{\prime}_{\ell}(a_{i}^{*})=a_{i}^{*}z_{\ell}$ for $3\leqslant i\leqslant 4$. Since $\zeta_{0}-\zeta_{0}^{\prime}$ is equal to $b_{K}^{1}(e_{3}+2e_{5})$, and $\zeta_{\ell}-\zeta_{\ell}^{\prime}=(\zeta_{0}-\zeta_{0}^{\prime})\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}z_{\ell}$ is also a coboundary, $\zeta_{\ell}$ and $\zeta_{\ell}^{\prime}$ represent the same cohomology class for $\ell\in\left\{0,6,8\right\}$. Moreover, as in types A and D, the elements $\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{j}$ form a basis of $\mathop{\rm HK}\nolimits^{2}(A)$.

If $\mathop{\rm char}\nolimits(\mathbb{F})\in\left\{2,3\right\}$, we need only prove that the extra elements are not coboundaries by fact 3.

If $\mathop{\rm char}\nolimits(\mathbb{F})=2$, we have $z_{6}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}\rho_{3}-\zeta_{8}=b_{k}^{1}(g)$ where $g$ is defined by $g(e_{2})=a_{2}^{*}c_{3}c_{0}c_{3}a_{2}$, and $\zeta_{8}$ is not a coboundary, therefore $\rho_{3}$ cannot be a coboundary. Moreover, assume that $\gamma_{4}=b_{K}^{1}(g^{\prime})$ is a coboundary. Then $g^{\prime}$ would be of coefficient weight $3$, and we would necessarily take values in $A_{3}=E\oplus\kappa(E)$ where $E$ is spanned by $c_{3}a_{0}$, $c_{0}a_{2},$ $c_{3}a_{2}$, $a_{3}c_{0},$ $a_{3}c_{3}$. This leads to a contradiction.

If $\mathop{\rm char}\nolimits(\mathbb{F})=3$, assume that $\rho_{5}$ is a coboundary $b_{k}^{1}(h),$ then $h$ is of weight $4$, and necessarily $h(e_{0})=\lambda_{0}a_{0}^{*}c_{3}a_{0}$, $h(e_{2})=\lambda_{2}a_{2}^{*}c_{3}a_{2}$ and $h(e_{3})=\lambda_{3}c_{3}c_{0}+\lambda_{3}^{\prime}c_{3}^{2}+\lambda_{3}^{\prime\prime}c_{0}c^{3}$, and by considering $b_{K}^{1}(h)(a_{0})$, $b_{K}^{1}(h)(a_{0}^{*})$, $b_{K}^{1}(h)(a_{2})$ and $b_{K}^{1}(h)(a_{2}^{*})$ we get a contradiction. Finally, the fact that $\gamma_{6}$ is not a coboundary follows from Lemma 6.3.1.

Koszul homology follows using duality (Theorem 4.3), as in Corollary 6.1.1.

6.3.2 Cup and cap products

We now determine the cup products of the elements in the bases of the Koszul cohomology spaces given above.

**Proposition **\theDf

Let $A$ be a preprojective algebra of type E6. Up to graded commutativity, the non zero cup products of elements in $\mathop{\rm HK}\nolimits^{\bullet}(A)$ are:

[TABLE]

Proof 6.8**.**

The first two cup products are clear.

For the cup products of $z_{6}$ with the $\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{i}$ and for $\mkern 2.0mu\overline{\mkern-2.0mu\zeta\mkern 1.0mu}\mkern-1.0mu_{0}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}\mkern 2.0mu\overline{\mkern-2.0mu\rho\mkern 1.0mu}\mkern-1.0mu_{5}$, we use Lemma 6.3.1.

The last cup product follows from the fact that we have $\zeta_{0}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}\rho_{3}-\gamma_{4}=b_{K}^{1}(g^{\prime})$ where $g^{\prime}(a_{0}^{*})=a_{0}^{*}c_{3}$ and $g^{\prime}(a_{3})=a_{3}c_{3}$. The cup product $z_{6}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}\mkern 2.0mu\overline{\mkern-2.0mu\rho\mkern 1.0mu}\mkern-1.0mu_{3}$ was already in the proof of Proposition 6.3.1.

Consideration of the coefficient weights yields the vanishing of the other cup products.

The cap products follow using duality, as in Corollary 6.1.2.

6.3.3 Higher Koszul cohomology and homology

As in type A, the differential $\partial_{\smile}^{1}$ sends $z_{\ell}$ to $2\mkern 2.0mu\overline{\mkern-2.0mu\zeta\mkern 1.0mu}\mkern-1.0mu_{\ell}$ for $\ell\in\left\{0,6,8\right\}$ and the differential $\partial_{\smile}^{2}$ is zero.

We then have the following higher Koszul cohomology.

**Proposition **\theDf

Let $A$ be a preprojective algebra of type E6.

If $\mathop{\rm char}\nolimits(\mathbb{F})=2$, then $\mathop{\rm HK}\nolimits^{\bullet}_{hi}(A)=\mathop{\rm HK}\nolimits^{\bullet}(A).$

If $\mathop{\rm char}\nolimits(\mathbb{F})=3$, then

[TABLE]

If $\mathop{\rm char}\nolimits(\mathbb{F})\not\in\left\{2,3\right\}$, then

[TABLE]

Higher Koszul homology follows from Theorem 4.3.

6.4 Koszul calculus for preprojective algebras of type E7

The preprojective algebra $A$ of type $E_{7}$ is defined by the quiver

[TABLE]

subject to the relations

[TABLE]

The Nakayama automorphism is given by $\nu(a_{i})=-a_{i}$ and $\nu(a_{i}^{*})=a_{i}^{*}$ for $i\in Q_{0}$.

To simplify notation, we shall denote by $c_{0}=a_{0}a_{0}^{*}$, $c_{2}=a_{2}a_{2}^{*}$ and $c_{3}=a_{3}^{*}a_{3}$ the three $2$-cycles at the vertex $3$.

The socle of $A$ is the part of weight $16$ of $A$. A basis of the socle is given by $\pi_{0}=(a_{0}^{*}c_{3}a_{0})^{4}$, $\pi_{1}=-a_{1}^{*}a_{2}^{*}c_{0}c_{3}(c_{3}c_{0})^{2}a_{2}a_{1}$, $\pi_{2}=-(a_{2}^{*}c_{0}a_{2})^{4}$, $\pi_{3}=(c_{3}c_{0})^{3}c_{3}^{2}$, $\pi_{4}=-(a_{3}c_{0}a_{3}^{*})^{4}$, $\pi_{5}=-a_{4}a_{3}(c_{3}c_{0})^{3}a_{3}^{*}a_{4}^{*}$ and $\pi_{6}=-a_{5}a_{4}(a_{3}c_{0}a_{3}^{*})^{3}a_{4}^{*}a_{5}^{*}$.

6.4.1 The Koszul cohomology and homology spaces in type E7

We define the following elements

• •

in $A$: $z_{0}=1$, $z_{8}=a_{0}^{*}c_{2}c_{0}c_{2}a_{0}-a_{2}^{*}c_{2}c_{0}c_{2}a_{2}-c_{2}c_{3}^{2}c_{2}+a_{3}c_{0}c_{2}c_{0}a_{3}^{*}-a_{4}a_{3}c_{2}^{2}a_{3}^{*}a_{4}^{*}+a_{5}a_{4}a_{3}c_{0}a_{3}^{*}a_{4}^{*}a_{5}^{*}$ and $z_{12}=a_{0}^{*}(c_{2}c_{0})^{2}c_{2}a_{0}^{*}+a_{2}^{*}(c_{0}c_{2})^{2}c_{0}a_{2}-(c_{3}c_{0}c_{3})^{2}+a_{3}c_{3}(c_{0}c_{3})^{2}a_{3}^{*}$;

• •

in $\mathop{\rm Hom}\nolimits_{k^{e}}(V,A)$: the maps $\zeta_{\ell}$ defined by $\zeta_{\ell}(a_{i})=a_{i}z_{\ell}$ for $\ell\in\left\{0,8,12\right\}$, the map $\rho_{3}$ defined by $\rho_{3}(a_{2})=c_{0}a_{2}$, $\rho_{3}(a_{3})=a_{3}c_{3}$, $\rho_{3}(a_{4})=a_{4}a_{3}a_{3}^{*}$ and $\rho_{3}(a_{2}^{*})=a_{2}^{*}c_{3}$, the map $\rho_{7}$ defined by $\rho_{7}(a_{0})=c_{3}^{3}a_{0}+c_{3}c_{0}c_{3}a_{0}$, $\rho_{7}(a_{3})=a_{3}c_{3}c_{0}c_{3}$ and $\rho_{7}(a_{3}^{*})=c_{3}c_{0}c_{3}a_{3}^{*}$, the map $\rho_{15}$ defined by $\rho_{15}(a_{0})=(c_{2}c_{0})^{3}c_{2}a_{0}$ and $\rho_{15}(a_{0}^{*})=a_{0}^{*}c_{2}(c_{0}c_{2})^{3}$ and the map $\rho_{5}$ defined by $\rho_{5}(a_{0})=-c_{2}c_{3}a_{0}$, $\rho_{5}(a_{2})=c_{3}c_{0}a_{2}$, $\rho_{5}(a_{3})=a_{3}c_{3}^{2}$, $\rho_{5}(a_{0}^{*})=a_{0}^{*}c_{3}^{2}$ and $\rho_{5}(a_{2}^{*})=-a_{2}^{*}c_{2}c_{0}$;

• •

in $\mathop{\rm Hom}\nolimits_{k^{e}}(R,A)$: the maps $h_{j}$ defined for $0\leqslant j\leqslant 6$ by $h_{j}(\sigma_{i})=\delta_{ij}e_{j}$ for all $i$, the map $\gamma_{4}$ defined by $\gamma_{4}(\sigma_{0})=a_{0}^{*}c_{3}a_{0}$, the map $\gamma_{8}$ defined by $\gamma_{8}(\sigma_{0})=a_{0}^{*}c_{3}^{3}a_{0}$, the map $\gamma_{16}$ defined by $\gamma_{16}(\sigma_{0})=\pi_{0}$ and the map and $\gamma_{6}$ defined by $\gamma_{6}(\sigma_{0})=a_{0}^{*}c_{3}^{2}a_{0}$.

**Lemma **\theDf

First assume that $\mathop{\rm char}\nolimits(\mathbb{F})=2$.

1. \edefitit(i)

Let $u_{16}\in\mathop{\rm Hom}\nolimits_{k^{e}}(R,A)$ be an element of weight $16$ so that $u_{16}(\sigma_{i})=\lambda_{i}\pi_{i}$ for $i\in Q_{0}$. Then $u_{16}$ is a coboundary if, and only if, $\sum_{i=0}^{6}{\lambda_{i}}=0$. 2. \edefitit(ii)

Let $u_{8}\in\mathop{\rm Hom}\nolimits_{k^{e}}(R,A)$ be an element of weight $8$ so that $u_{8}(\sigma_{0})=\lambda_{0}a_{0}^{*}c_{3}^{3}a_{0}$, $u_{8}(\sigma_{2})=\lambda_{2}a_{2}^{*}c_{3}c_{0}c_{3}a_{2}+\lambda_{2}^{\prime}a_{2}^{*}c_{3}^{2}c_{0}a_{2}$, $u_{8}(\sigma_{3})=\lambda_{3}c_{3}^{3}c_{0}+\lambda_{3}^{\prime}c_{c}c_{3}^{3}+\lambda_{3}^{\prime\prime}c_{0}c_{3}^{2}c_{0}+\lambda_{3}^{\prime\prime\prime}c_{3}^{2}c_{0}c_{3}$, $u_{8}(\sigma_{4})=\lambda_{4}a_{3}c_{3}^{2}c_{0}a_{3}^{*}+\lambda_{4}^{\prime}a_{3}c_{3}c_{0}c_{3}a_{3}^{*}+\lambda_{4}^{\prime\prime}a_{3}c_{0}c_{3}^{2}a_{3}^{*}$, $u_{8}(\sigma_{5})=\lambda_{5}a_{4}a_{3}c_{3}c_{0}a_{3}^{*}a_{4}^{*}+\lambda_{5}^{\prime}a_{4}a_{3}c_{0}c_{3}a_{3}^{*}a_{4}^{*}$ and $u_{8}(\sigma_{6})=a_{5}a_{4}a_{3}a_{0}a_{0}^{*}a_{3}^{*}a_{4}^{*}a_{5}^{*}$. Then $u_{8}$ is a coboundary if, and only if, $\lambda_{0}+\lambda_{2}+\lambda_{2}^{\prime}+\lambda_{3}+\lambda_{3}^{\prime}+\lambda_{3}^{\prime\prime\prime}+\lambda_{4}+\lambda_{4}^{\prime}+\lambda_{4}^{\prime\prime}+\lambda_{5}+\lambda_{5}^{\prime}+\lambda_{6}=0$.

Now assume that $\mathop{\rm char}\nolimits(\mathbb{F})=3$.

1. (iii)

Let $u_{6}\in\mathop{\rm Hom}\nolimits_{k^{e}}(R,A)$ be an element of weight $6$ so that $u_{6}(\sigma_{0})=\lambda_{0}a_{0}^{*}c_{3}^{2}a_{0}$, $u_{6}(\sigma_{1})=\lambda_{1}a_{1}^{*}a_{2}^{*}c_{0}a_{2}^{*}a_{1}^{*}$, $u_{6}(\sigma_{2})=\lambda_{2}a_{2}^{*}c_{3}c_{0}a_{2}+\lambda_{2}^{\prime}a_{2}^{*}c_{3}^{2}a_{2}$, $u_{6}(\sigma_{3})=\lambda_{3}c_{3}c_{0}c_{3}+\lambda_{3}^{\prime}c_{3}^{2}c_{0}+\lambda_{3}^{\prime\prime}c_{0}c_{3}c_{0}+\lambda_{3}^{\prime\prime\prime}c_{3}^{3}+$, $u_{6}(\sigma_{4})=\lambda_{4}a_{3}c_{0}c_{3}a_{3}^{*}+\lambda_{4}^{\prime}a_{3}c_{3}c_{0}a_{3}^{*}$ and $u_{6}(\sigma_{5})=\lambda_{5}a_{4}a_{3}c_{0}a_{3}^{*}a_{4}^{*}$. Then $u_{6}$ is a coboundary if, and only if, $\sum_{i=0}^{5}{\lambda_{i}}-\lambda_{2}^{\prime}+\lambda_{3}^{\prime}-\lambda_{4}^{\prime}=0$.

Proof 6.9**.**

For each $\ell\in\left\{16,6\right\}$, if the map $u_{\ell}$ were a coboundary, it would be the image of a map $g_{\ell}\in\mathop{\rm Hom}\nolimits_{k^{e}}(V,A)$ whose coefficients would be in the space generated by the paths between adjacent vertices with weight $\ell-1$. The proof is then straightforward once we know bases of these spaces. Note that once we have a basis of $\bigoplus_{\alpha\in Q_{1}}e_{\mathfrak{t}(\alpha)}A_{w}e_{\mathfrak{s}(\alpha)}$, applying $\kappa$ gives a basis of $e_{\mathfrak{s}(\alpha)}A_{w}e_{\mathfrak{t}(\alpha)}$ for a given weight $w$.

In weight $15$, a basis of $\bigoplus_{\alpha\in Q_{1}}e_{\mathfrak{t}(\alpha)}A_{15}e_{\mathfrak{s}(\alpha)}$ is given by $\mkern 2.0mu\overline{\mkern-2.0mua\mkern 1.0mu}\mkern-1.0mu_{0}=(a_{0}^{*}c_{3}a_{0})^{3}a_{0}^{*}c_{3}$, $\mkern 2.0mu\overline{\mkern-2.0mua\mkern 1.0mu}\mkern-1.0mu_{1}=-a_{1}^{*}a_{2}^{*}c_{0}c_{3}(c_{3}c_{0})^{2}a_{2}$, $\mkern 2.0mu\overline{\mkern-2.0mua\mkern 1.0mu}\mkern-1.0mu_{2}=-a_{2}^{*}c_{0}(c_{2}c_{0})^{3}$, $\mkern 2.0mu\overline{\mkern-2.0mua\mkern 1.0mu}\mkern-1.0mu_{3}=(c_{3}c_{0})^{3}c_{3}a_{3}^{*}$, $\mkern 2.0mu\overline{\mkern-2.0mua\mkern 1.0mu}\mkern-1.0mu_{4}=-a_{3}(c_{3}c_{0})^{3}a_{3}^{*}a_{4}^{*}$, $\mkern 2.0mu\overline{\mkern-2.0mua\mkern 1.0mu}\mkern-1.0mu_{5}=a_{4}(a_{3}c_{0}a_{3}^{*})^{3}a_{4}^{*}a_{5}^{*}$.

In weight $7$, a basis of $\bigoplus_{\alpha\in Q_{1}}e_{\mathfrak{t}(\alpha)}A_{7}e_{\mathfrak{s}(\alpha)}$ is given by $c_{3}^{3}a_{0}$, $c_{3}c_{0}c_{3}a_{0}$, $a_{2}^{*}c_{3}^{2}a_{2}a_{1}$, $c_{3}^{2}c_{0}a_{2}$, $c_{0}c_{3}^{2}a_{2}$, $c_{3}c_{0}c_{3}a_{2}$, $a_{3}c_{3}c_{0}c_{3}$, $a_{3}c_{0}c_{3}^{2}$, $a_{3}c_{3}^{2}c_{0}$. $a_{4}a_{3}c_{3}c_{0}a_{3}^{*}$, $a_{4}a_{3}c_{0}c_{3}a_{3}^{*}$, $a_{5}a_{4}a_{3}c_{0}a_{3}^{*}a_{4}^{*}$.

In weight $5$, a basis of $\bigoplus_{\alpha\in Q_{1}}e_{\mathfrak{t}(\alpha)}A_{5}e_{\mathfrak{s}(\alpha)}$ is given by $c_{3}^{2}a_{0}$, $c_{0}c_{3}a_{0}$, $a_{2}^{*}c_{0}a_{2}a_{1}$, $c_{3}c_{0}a_{2}$, $c_{0}c_{3}a_{2}$, $a_{3}c_{3}c_{0}$, $a_{3}c_{0}c_{3}$, $a_{3}c_{3}^{2}$, $a_{4}a_{3}c_{0}a_{3}^{*}$.

**Proposition **\theDf

Let $A$ be a preprojective algebra of type E7.

1. \edefitit(i)

The elements in $\left\{z_{0},z_{8},z_{12}\right\}\cup\left\{\pi_{i};i\in Q_{0}\right\}$ form a basis of $\mathop{\rm HK}\nolimits^{0}(A)$. 2. \edefitit(ii)

If $\mathop{\rm char}\nolimits(\mathbb{F})\not\in\left\{2,3\right\}$, the elements in $\left\{\mkern 2.0mu\overline{\mkern-2.0mu\zeta\mkern 1.0mu}\mkern-1.0mu_{\ell};\ell=0,8,12\right\}$ form a basis of $\mathop{\rm HK}\nolimits^{1}(A)$.

If $\mathop{\rm char}\nolimits(\mathbb{F})=2$, the elements in $\left\{\mkern 2.0mu\overline{\mkern-2.0mu\zeta\mkern 1.0mu}\mkern-1.0mu_{\ell};\ell=0,8,12\right\}\cup\left\{\mkern 2.0mu\overline{\mkern-2.0mu\rho\mkern 1.0mu}\mkern-1.0mu_{3},\mkern 2.0mu\overline{\mkern-2.0mu\rho\mkern 1.0mu}\mkern-1.0mu_{7},\mkern 2.0mu\overline{\mkern-2.0mu\rho\mkern 1.0mu}\mkern-1.0mu_{15}\right\}$ form a basis of $\mathop{\rm HK}\nolimits^{1}(A)$.

If $\mathop{\rm char}\nolimits(\mathbb{F})=3$, the elements in $\left\{\mkern 2.0mu\overline{\mkern-2.0mu\zeta\mkern 1.0mu}\mkern-1.0mu_{\ell};\ell=0,8,12\right\}\cup\left\{\mkern 2.0mu\overline{\mkern-2.0mu\rho\mkern 1.0mu}\mkern-1.0mu_{5}\right\}$ form a basis of $\mathop{\rm HK}\nolimits^{1}(A)$. 3. \edefitit(iii)

If $\mathop{\rm char}\nolimits(\mathbb{F})\not\in\left\{2,3\right\}$, the elements in $\left\{\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{j};j\in Q_{0}\right\}$ form a basis of $\mathop{\rm HK}\nolimits^{2}(A)$.

If $\mathop{\rm char}\nolimits(\mathbb{F})=2$, the elements in $\left\{\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{j};j\in Q_{0}\right\}\cup\left\{\mkern 2.0mu\overline{\mkern-2.0mu\gamma\mkern 1.0mu}\mkern-1.0mu_{4},\mkern 2.0mu\overline{\mkern-2.0mu\gamma\mkern 1.0mu}\mkern-1.0mu_{8},\mkern 2.0mu\overline{\mkern-2.0mu\gamma\mkern 1.0mu}\mkern-1.0mu_{16}\right\}$ form a basis of $\mathop{\rm HK}\nolimits^{2}(A)$.

If $\mathop{\rm char}\nolimits(\mathbb{F})=3$, the elements in $\left\{\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{j};j\in Q_{0}\right\}\cup\left\{\mkern 2.0mu\overline{\mkern-2.0mu\gamma\mkern 1.0mu}\mkern-1.0mu_{6}\right\}$ form a basis of $\mathop{\rm HK}\nolimits^{2}(A)$.

Proof 6.10**.**

The centre was given in [22], so we have (i).

For $\mathop{\rm HK}\nolimits^{1}(A)$ and $\mathop{\rm HK}\nolimits^{2}(A)$, the number of elements in the statement is equal to the dimension of the corresponding cohomology space. Moreover, all the elements in the statement are indeed cocycles.

If $\mathop{\rm char}\nolimits(\mathbb{F})$ is not $2$ or $3$, a basis of $\mathop{\rm HK}\nolimits^{1}(A)=\mathop{\rm HH}\nolimits^{1}(A)$ was given in [22]. It consists of the classes of the $\zeta^{\prime}_{\ell}$ with $\ell\in\left\{0,8,12\right\}$ where $\zeta^{\prime}_{\ell}(a_{i})=a_{i}z_{\ell}$ for $0\leqslant i\leqslant 2$ and $\zeta^{\prime}_{\ell}(a_{i}^{*})=a_{i}^{*}z_{\ell}$ for $3\leqslant i\leqslant 5$. Since $\zeta_{0}-\zeta_{0}^{\prime}$ is equal to $b_{K}^{1}(e_{3}+2e_{5}+3e_{6})$, and $\zeta_{\ell}-\zeta_{\ell}^{\prime}=(\zeta_{0}-\zeta_{0}^{\prime})\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}z_{\ell}$ is also a coboundary, $\zeta_{\ell}$ and $\zeta_{\ell}^{\prime}$ represent the same cohomology class for $\ell\in\left\{0,8,12\right\}$. Moreover, as in types A, D and E6, the elements $\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{j}$ form a basis of $\mathop{\rm HK}\nolimits^{2}(A)$.

If $\mathop{\rm char}\nolimits(\mathbb{F})\in\left\{2,3\right\}$, we need only prove that the extra elements are not coboundaries by fact 3.

If $\mathop{\rm char}\nolimits(\mathbb{F})=2$, it follows from Lemma 6.4.1 that $\zeta_{0}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}\rho_{15}-\gamma_{16}$, $\zeta_{8}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}\rho_{7}-\gamma_{16}$, $\zeta_{12}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}\rho_{3}-\gamma_{16}$, $z_{8}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}\gamma_{8}-\gamma_{16}$ and $z_{12}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}\gamma_{4}-\gamma_{16}$ are coboundaries. Therefore it is enough to check that $\gamma_{16}$ is not a coboundary, and this also follows from Lemma 6.4.1.

If $\mathop{\rm char}\nolimits(\mathbb{F})=3$, again using Lemma 6.4.1, $\zeta_{0}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}\rho_{5}-\gamma_{6}$ is a coboundary and $\gamma_{6}$ is not a coboundary, therefore $\rho_{5}$ is not a coboundary either.

Koszul homology follows using duality (Theorem 4.3), as in Corollary 6.1.1.

6.4.2 Cup and cap products

We now determine the cup products of the elements in the bases of the Koszul cohomology spaces given above.

**Proposition **\theDf

Let $A$ be a preprojective algebra of type E7. Up to graded commutativity, the non zero cup products of elements in $\mathop{\rm HK}\nolimits^{\bullet}(A)$ are:

[TABLE]

Proof 6.11**.**

Most of the cup-products are easy to compute, follow from Lemma 6.4.1 or vanish for weight reasons. The remaining ones are obtained as follows (at the level of cochains):

[TABLE]

where $h\in\mathop{\rm Hom}\nolimits_{k^{e}}(V,A_{3})$ is defined by $h(a_{0})=c_{3}a_{0}$ and $h(a_{2})=c_{2}a_{2}$.

The cap products follow using duality, as in Corollary 6.1.2.

6.4.3 Higher Koszul cohomology and homology

As in types A, D and E6, the differential $\partial_{\smile}^{1}$ sends $z_{\ell}$ to $2\mkern 2.0mu\overline{\mkern-2.0mu\zeta\mkern 1.0mu}\mkern-1.0mu_{\ell}$ for $\ell\in\left\{0,8,12\right\}$ and the differential $\partial_{\smile}^{2}$ is zero except when $\mathop{\rm char}\nolimits(\mathbb{F})=3$ where $\partial_{\smile}^{2}(\mkern 2.0mu\overline{\mkern-2.0mu\rho\mkern 1.0mu}\mkern-1.0mu_{5})=\mkern 2.0mu\overline{\mkern-2.0mu\gamma\mkern 1.0mu}\mkern-1.0mu_{6}$.

We then have the following higher Koszul cohomology.

**Proposition **\theDf

Let $A$ be a preprojective algebra of type E7.

If $\mathop{\rm char}\nolimits(\mathbb{F})=2$, then $\mathop{\rm HK}\nolimits^{\bullet}_{hi}(A)=\mathop{\rm HK}\nolimits^{\bullet}(A).$

If $\mathop{\rm char}\nolimits(\mathbb{F})\neq 2$, then

[TABLE]

Higher Koszul homology follows from Theorem 4.3.

6.5 Koszul calculus for preprojective algebras of type E8

The preprojective algebra $A$ of type $E_{8}$ is defined by the quiver

[TABLE]

subject to the relations

[TABLE]

The Nakayama automorphism is given by $\nu(a_{i})=-a_{i}$ and $\nu(a_{i}^{*})=a_{i}^{*}$ for $i\in Q_{0}$.

To simplify notation, we shall denote by $c_{0}=a_{0}a_{0}^{*}$, $c_{2}=a_{2}a_{2}^{*}$ and $c_{3}=a_{3}^{*}a_{3}$ the three $2$-cycles at the vertex $3$.

The socle of $A$ is the part of weight $28$ of $A$. A basis of the socle is given by $\pi_{0}=(a_{0}^{*}c_{2}a_{0})^{7}$, $\pi_{1}=a_{1}^{*}a_{2}^{*}(c_{0}c_{2})^{5}c_{2}c_{0}a_{2}a_{1}$, $\pi_{2}=-(a_{2}^{*}c_{0}a_{2})^{7}$, $\pi_{3}=(c_{2}c_{0})^{7}$, $\pi_{4}=-a_{3}(c_{2}c_{0})^{6}c_{2}a_{3}^{*}$, $\pi_{5}=a_{4}a_{3}(c_{2}c_{0})^{6}a_{3}^{*}a_{4}^{*}$, $\pi_{6}=a_{5}a_{4}a_{3}(c_{0}c_{2})^{5}c_{0}a_{3}^{*}a_{4}^{*}a_{5}^{*}$ and $\pi_{7}=-a_{6}a_{5}a_{4}a_{3}(c_{0}c_{2})^{3}(c_{2}c_{0})^{2}a_{3}^{*}a_{4}^{*}a_{5}^{*}a_{6}^{*}$.

6.5.1 The Koszul cohomology and homology spaces in type E8

We define the following elements

1. $\bullet$

in $A$:

1. ✦

$z_{0}=1$, 2. ✦

$z_{12}=a_{5}a_{4}a_{3}c_{0}c_{2}c_{0}a_{3}^{*}a_{4}^{*}a_{5}^{*}+a_{4}a_{3}(c_{2}c_{0})^{2}a_{3}^{*}a_{4}^{*}+a_{3}(c_{2}c_{0})^{2}c_{2}a_{3}^{*}-(c_{3}c_{2}c_{3})^{2}+a_{2}^{*}(c_{0}c_{2})^{2}c_{0}a_{2}-a_{1}^{*}a_{2}^{*}c_{0}c_{2}^{2}c_{0}a_{2}a_{1}+a_{0}^{*}(c_{2}c_{0})^{2}c_{2}a_{0}$, 3. ✦

$z_{20}=a_{5}a_{4}a_{3}(c_{0}c_{2})^{3}c_{0}a_{3}^{*}a_{4}^{*}a_{5}^{*}+a_{4}a_{3}(c_{0}c_{2})^{2}(c_{2}c_{0})^{2}a_{3}^{*}a_{4}^{*}+a_{3}c_{0}(c_{2}c_{0})^{4}a_{3}^{*}+(c_{2}c_{0})^{5}-(c_{0}c_{2}^{2})^{3}c_{0}+(c_{0}c_{2})^{5}+a_{2}^{*}(c_{2}c_{0}c_{2})^{3}a_{2}+a_{0}^{*}(c_{2}c_{0})^{4}c_{2}a_{0}$ and 4. ✦

$z_{24}=z_{12}^{2}$; 2. $\bullet$

in $\mathop{\rm Hom}\nolimits_{k^{e}}(V,A)$:

1. ✦

the maps $\zeta_{\ell}$ defined by $\zeta_{\ell}(a_{i})=a_{i}z_{\ell}$ for $\ell\in\left\{0,12,20,24\right\}$, 2. ✦

the map $\rho_{3}$ defined by $\rho_{3}(a_{2})=c_{0}a_{2}$, $\rho_{3}(a_{3})=a_{3}c_{3}$, $\rho_{3}(a_{4})=a_{4}a_{3}a_{3}^{*}$, $\rho_{3}(a_{5})=a_{5}a_{4}a_{4}^{*}$ and $\rho_{3}(a_{2}^{*})=a_{2}^{*}c_{3}$, 3. ✦

the map $\rho_{7}$ defined by $\rho_{7}(a_{0})=c_{0}c_{3}^{2}a_{0}$, $\rho_{7}(a_{3})=a_{3}c_{0}c_{3}^{2}+a_{3}c_{3}^{2}c_{0}+a_{3}c_{0}c_{3}c_{0}$ and $\rho_{7}(a_{3}^{*})=c_{3}c_{0}c_{3}a_{3}^{*}$, 4. ✦

the map $\rho_{15}$ defined by $\rho_{15}(a_{3})=a_{3}c_{2}(c_{2}c_{0})^{2}$, $\rho_{15}(a_{4})=a_{4}a_{3}(c_{0}c_{3})^{2}a_{3}^{*}+a_{4}a_{3}(c_{0}c_{3}^{2})^{2}a_{3}^{*}+a_{4}a_{3}(c_{3}c_{0})^{3}a_{3}^{*}$, $\rho_{15}(a_{5})=a_{5}a_{4}a_{3}c_{0}(c_{3}c_{0})^{2}a_{3}^{*}a_{4}^{*}$, $\rho_{15}(a_{0}^{*})=a_{0}^{*}c_{3}(c_{3}c_{0})^{3}$ and $\rho_{15}(a_{5}^{*})=a_{4}a_{3}(c_{0}c_{3})^{2}c_{0}a_{3}^{*}a_{4}^{*}a_{5}^{*}$, 5. ✦

the map $\rho_{27}$ defined by $\rho_{27}(a_{3})=a_{3}c_{0}(c_{3}c_{0})^{6}$ and $\rho_{27}(a_{3}^{*})=(c_{3}c_{0})^{6}c_{3}a_{3}^{*}$, 6. ✦

the map $\rho_{5}$ defined by $\rho_{5}(a_{0})=-c_{0}c_{3}a_{0}-c_{3}^{2}a_{0}$, $\rho_{5}(a_{3})=a_{3}c_{3}c_{0}+a_{3}c_{3}^{2}$, $\rho_{5}(a_{4})=a_{4}a_{3}c_{3}a_{3}^{*}$, $\rho_{5}(a_{0}^{*})=a_{0}^{*}c_{3}^{2}$ and $\rho_{5}(a_{3}^{*})=-c_{3}c_{0}a_{3}^{*}$, 7. ✦

the map $\rho_{17}$ defined by defined by $\rho_{17}(a_{0})=-c_{3}^{2}(c_{0}c_{3})^{3}a_{0}-(c_{0}c_{3})^{4}a_{0}$, $\rho_{17}(a_{3})=a_{3}(c_{3}c_{0})^{4}+a_{3}(c_{0}c_{3})^{4}+a_{3}(c_{3}c_{0})^{3}c_{3}^{2}$, $\rho_{17}(a_{0}^{*})=a_{0}^{*}(c_{2}c_{0})^{3}c_{3}^{2}$ and $\rho_{17}(a_{3}^{*})=-(c_{3}c_{0})^{4}a_{3}^{*}$, and 8. ✦

the map $\rho_{9}$ defined by $\rho_{9}(a_{0})=-2c_{3}^{2}c_{0}c_{3}a_{0}+2c_{3}c_{0}c_{3}^{2}a_{0}+(c_{0}c_{3})^{2}a_{0}$, $\rho_{9}(a_{2})=c_{2}^{2}c_{0}c_{2}a_{2}+(c_{2}c_{0})^{2}a_{2}$, $\rho_{9}(a_{3})=-a_{3}(c_{0}c_{3})^{2}$, $\rho_{9}(a_{0}^{*})=2a_{0}^{*}c_{3}^{2}c_{0}c_{3}-2a_{0}^{*}c_{3}c_{0}c_{3}^{2}+a_{0}^{*}(c_{3}c_{0})^{2}$, $\rho_{9}(a_{2}^{*})=a_{2}^{*}c_{2}c_{0}c_{2}^{2}-a_{2}^{*}(c_{2}c_{0})^{2}$ and $\rho_{9}(a_{3}^{*})=(c_{0}c_{3})^{2}a_{3}^{*}$; 3. $\bullet$

in $\mathop{\rm Hom}\nolimits_{k^{e}}(R,A)$: the maps $h_{j}$ defined for $0\leqslant j\leqslant 7$ by $h_{j}(\sigma_{i})=\delta_{ij}e_{j}$ for all $i$, and

[TABLE]

**Lemma **\theDf

First assume that $\mathop{\rm char}\nolimits(\mathbb{F})=2$.

1. \edefitit(i)

Let $u_{28}\in\mathop{\rm Hom}\nolimits_{k^{e}}(R,A)$ be an element of weight $28$, so that $u_{28}(\sigma_{i})=\lambda_{i}\pi_{i}$ for all $i\in Q_{0}$. Then $u_{28}$ is a coboundary if, and only if, $\sum_{i\in Q_{0}}\lambda_{i}=0$.

Now assume that $\mathop{\rm char}\nolimits(\mathbb{F})=3$.

1. (ii)

Let $u_{18}\in\mathop{\rm Hom}\nolimits_{k^{e}}(R,A)$ be an element of weight $18$, so that $u_{18}(\sigma_{0})=\lambda_{0}a_{0}^{*}c_{3}^{2}(c_{0}c_{3})^{3}a_{0}+\lambda_{0}^{\prime}a_{0}^{*}c_{3}c_{0}(c_{3}c_{0}c_{3})^{2}a_{0}$, $u_{18}(\sigma_{1})=\lambda_{1}a_{1}^{*}a_{2}^{*}c_{0}(c_{2}c_{0})^{3}a_{2}a_{1}$, $u_{18}(\sigma_{2})=\lambda_{2}a_{2}^{*}(c_{2}c_{0}c_{2})^{2}c_{0}c_{2}a_{2}+\lambda_{2}^{\prime}a_{2}^{*}(c_{0}c_{2})^{4}a_{2}+\lambda_{2}^{\prime\prime}a_{2}^{*}(c_{2}c_{0})^{4}a_{2}$, $u_{18}(\sigma_{3})=\lambda_{3}c_{0}(c_{3}c_{0})^{4}+\lambda_{3}^{\prime}(c_{3}c_{0})^{4}c_{3}+\lambda_{3}^{\prime\prime}(c_{3}c_{0})^{2}c_{3}(c_{3}c_{0})^{2}+\lambda_{3}^{(3)}(c_{0}c_{3})^{2}c_{3}(c_{0}c_{3})^{2}+\lambda_{3}^{(4)}(c_{3}c_{0})^{3}c_{3}^{2}c_{0}+\lambda_{3}^{(5)}c_{0}c_{3}^{2}(c_{0}c_{3})^{3}$, $u_{18}(\sigma_{4})=\lambda_{4}a_{3}(c_{3}c_{0})^{4}a_{3}^{*}+\lambda_{4}^{\prime}a_{3}(c_{0}c_{3})^{4}a_{3}^{*}+\lambda_{4}^{\prime\prime}a_{3}c_{3}^{2}(c_{3}c_{0})^{3}a_{3}^{*}+\lambda_{4}^{(3)}a_{3}(c_{0}c_{3})^{3}c_{3}^{2}a_{3}^{*}$, $u_{18}(\sigma_{5})=\lambda_{5}a_{4}a_{3}(c_{3}c_{0})^{3}c_{3}a_{3}^{*}a_{4}^{*}+\lambda_{5}^{\prime}a_{4}a_{3}c_{3}(c_{3}c_{0})^{3}a_{3}^{*}a_{4}^{*}+\lambda_{5}^{\prime\prime}a_{4}a_{3}(c_{0}c_{3})^{3}c_{3}a_{3}^{*}a_{4}^{*}$, $u_{18}(\sigma_{6})=\lambda_{6}a_{5}a_{4}a_{3}(c_{3}c_{0})^{3}a_{3}^{*}a_{4}^{*}a_{5}^{*}+\lambda_{6}^{\prime}a_{5}a_{4}a_{3}(c_{0}c_{3})^{3}a_{3}^{*}a_{4}^{*}a_{5}^{*}$, $u_{18}(\sigma_{7})=\lambda_{7}a_{6}a_{5}a_{4}a_{3}(c_{0}c_{3})^{2}c_{0}a_{3}^{*}a_{4}^{*}a_{5}^{*}a_{6}^{*}$. Then $u_{18}$ is a coboundary if, and only if, $\lambda_{0}+\lambda_{0}^{\prime}+\lambda_{1}+\lambda_{2}^{\prime}+\lambda_{2}^{\prime\prime}+\lambda_{3}^{\prime}+\lambda_{3}^{\prime\prime}+\lambda_{3}^{(3)}+\lambda_{3}^{(4)}+\lambda_{3}^{(5)}+\lambda_{4}+\lambda_{4}^{\prime}-\lambda_{4}^{\prime\prime}-\lambda_{4}^{(3)}-\lambda_{5}-\lambda_{5}^{\prime}-\lambda_{5}^{\prime\prime}-\lambda_{6}-\lambda_{6}^{\prime}-\lambda_{7}=0$. 2. (iii)

The map $\rho_{17}\in\mathop{\rm Hom}\nolimits_{k^{e}}(V,A)$ is not a coboundary.

Now assume that $\mathop{\rm char}\nolimits(\mathbb{F})=5$.

1. (iv)

Let $u_{10}\in\mathop{\rm Hom}\nolimits_{k^{e}}(R,A)$ be an element of weight $10$, so that $u_{10}(\sigma_{0})=\lambda_{0}a_{0}^{*}c_{3}^{2}c_{0}c_{3}a_{0}+\lambda_{0}^{\prime}a_{0}^{*}c_{3}c_{0}c_{3}^{2}a_{0}$, $u_{10}(\sigma_{1})=\lambda_{1}a_{1}^{*}a_{2}^{*}c_{0}c_{2}c_{0}a_{2}a_{1}$, $u_{10}(\sigma_{2})=\lambda_{2}a_{2}^{*}(c_{0}c_{2})^{2}a_{2}+\lambda_{2}^{\prime}a_{2}^{*}(c_{2}c_{0})^{2}a_{2}+\lambda_{2}^{\prime\prime}a_{2}^{*}c_{0}c_{2}^{2}c_{0}a_{2}$, $u_{10}(\sigma_{3})=\lambda_{3}c_{0}(c_{3}c_{0})^{2}+\lambda_{3}^{\prime}c_{3}(c_{0}c_{3})^{2}+\lambda_{3}^{\prime\prime}c_{3}(c_{3}c_{0})^{2}+\lambda_{3}^{(3)}(c_{0}c_{3})^{2}c_{3}+\lambda_{3}{(4)}c_{3}c_{0}c_{3}^{2}c_{0}+\lambda_{3}^{(5)}c_{0}c_{3}^{2}c_{0}c_{3}$, $u_{10}(\sigma_{4})=\lambda_{4}a_{3}(c_{0}c_{3})^{2}a_{3}^{*}+\lambda_{4}^{\prime}a_{3}(c_{3}c_{0})^{2}a_{3}^{*}+\lambda_{4}^{\prime\prime}a_{3}c_{3}c_{0}c_{3}^{2}a_{3}^{*}+\lambda_{4}^{(3)}a_{3}c_{3}^{2}c_{0}c_{3}a_{3}^{*}$, $u_{10}(\sigma_{5})=\lambda_{5}a_{4}a_{3}c_{0}c_{3}c_{0}a_{3}^{*}a_{4}^{*}+\lambda_{5}^{\prime}a_{4}a_{3}c_{3}^{2}c_{0}a_{3}^{*}a_{4}^{*}+\lambda_{5}^{\prime\prime}a_{4}a_{3}c_{0}c_{3}^{2}a_{3}^{*}a_{4}^{*}$, $u_{10}(\sigma_{6})=\lambda_{6}a_{5}a_{4}a_{3}c_{3}c_{0}a_{3}^{*}a_{4}^{*}a_{5}^{*}+\lambda_{6}^{\prime}a_{5}a_{4}a_{3}c_{0}c_{3}a_{3}^{*}a_{4}^{*}a_{5}^{*}$, $u_{10}(\sigma_{7})=\lambda_{7}a_{6}a_{5}a_{4}a_{3}c_{0}a_{3}^{*}a_{4}^{*}a_{5}^{*}a_{6}^{*}$. Then $u_{10}$ is a coboundary if, and only if, $\lambda_{0}+\lambda_{0}^{\prime}+\lambda_{1}+\lambda_{2}+\lambda_{2}^{\prime}+\lambda_{2}^{\prime\prime}+\lambda_{3}^{\prime}+\lambda_{3}^{\prime\prime}+\lambda_{3}^{(3)}+\lambda_{3}^{(4)}+\lambda_{3}^{(5)}+\lambda_{4}-\lambda_{4}^{\prime}+2\lambda_{4}^{\prime\prime}+2\lambda_{4}^{(3)}+\lambda_{5}+2\lambda_{5}^{\prime}+2\lambda_{5}^{\prime\prime}+2\lambda_{6}+2\lambda_{6}^{\prime}+2\lambda_{7}=0$.

Proof 6.12**.**

For each $\ell\in\left\{28,18,10\right\}$, if the map $u_{\ell}$ were a coboundary, it would be the image of a map $g_{\ell}\in\mathop{\rm Hom}\nolimits_{k^{e}}(V,A)$ whose coefficients would be in the space generated by the paths between adjacent vertices with weight $\ell-1$. The proof of (i), (ii) and (iv) is then straightforward once we know bases of these spaces.

In weight $27$, a basis is given by $\mkern 2.0mu\overline{\mkern-2.0mua\mkern 1.0mu}\mkern-1.0mu_{0}=a_{0}^{*}c_{2}(c_{0}c_{2})^{6}$, $\mkern 2.0mu\overline{\mkern-2.0mua\mkern 1.0mu}\mkern-1.0mu_{1}=a_{1}^{*}a_{2}^{*}(c_{0}c_{2})^{5}c_{3}^{2}a_{2}$, $\mkern 2.0mu\overline{\mkern-2.0mua\mkern 1.0mu}\mkern-1.0mu_{2}=-a_{2}^{*}c_{0}(c_{2}c_{0})^{6}$, $\mkern 2.0mu\overline{\mkern-2.0mua\mkern 1.0mu}\mkern-1.0mu_{3}=(c_{2}c_{0})^{6}c_{2}a_{3}^{*}$, $\mkern 2.0mu\overline{\mkern-2.0mua\mkern 1.0mu}\mkern-1.0mu_{4}=-a_{3}(c_{2}c_{0})^{6}a_{3}^{*}a_{4}^{*}$, $\mkern 2.0mu\overline{\mkern-2.0mua\mkern 1.0mu}\mkern-1.0mu_{5}=-a_{4}a_{3}c_{0}(c_{2}c_{0})^{5}a_{3}^{*}a_{4}^{*}a_{5}^{*}$, $\mkern 2.0mu\overline{\mkern-2.0mua\mkern 1.0mu}\mkern-1.0mu_{6}=a_{5}a_{4}a_{3}(c_{0}c_{2})^{3}(c_{2}c_{0})^{2}a_{3}^{*}a_{4}^{*}a_{5}^{*}a_{6}^{*}$, and the $\kappa(\mkern 2.0mu\overline{\mkern-2.0mua\mkern 1.0mu}\mkern-1.0mu_{i})$ for all $i\in Q_{0}$.

In weight $17$, the space $\bigoplus_{\alpha\in Q_{1}}e_{\mathfrak{t}(\alpha)}A_{18}e_{\mathfrak{s}(\alpha)}$ has basis $(c_{0}c_{3})^{4}a_{0}$, $c_{3}^{2}(c_{0}c_{3})^{3}a_{0}$, $c_{3}c_{0}(c_{3}c_{0}c_{3})^{2}a_{0}$, $a_{2}^{*}c_{2}c_{0}c_{2}(c_{2}c_{0})^{2}a_{2}a_{1}$, $a_{2}^{*}(c_{0}c_{2})^{3}c_{0}a_{2}a_{1}$, $(c_{3}c_{0})^{4}a_{2}$, $(c_{0}c_{3})^{4}a_{2}$, $(c_{0}c_{3})^{3}c_{3}c_{0}a_{2}$, $c_{3}c_{0}(c_{3}c_{0}c_{3})^{2}a_{2}$, $a_{3}(c_{3}c_{0})^{4}$, $a_{3}(c_{0}c_{3})^{4}$, $a_{3}(c_{0}c_{3})^{3}c_{3}c_{0}$, $a_{3}c_{0}c_{3}(c_{3}c_{0})^{3}$, $a_{3}(c_{3}c_{0})^{3}c_{3}^{2}$, $a_{4}a_{3}(c_{0}c_{3})^{3}c_{0}a_{3}^{*}$, $a_{4}a_{3}(c_{0}c_{3})^{3}c_{3}a_{3}^{*}$, $a_{4}a_{3}c_{3}(c_{3}c_{0})^{3}a_{3}^{*}$, $a_{5}a_{4}a_{3}(c_{3}c_{0})^{3}a_{3}^{*}a_{4}^{*}$, $a_{5}a_{4}a_{3}(c_{0}c_{3})^{3}a_{3}^{*}a_{4}^{*}$, $a_{6}a_{5}a_{4}a_{3}(c_{0}c_{3})^{2}c_{0}a_{3}^{*}a_{4}^{*}a_{5}^{*}$ and $\bigoplus_{\alpha\in Q_{1}}e_{\mathfrak{s}(\alpha)}A_{18}e_{\mathfrak{t}(\alpha)}=\kappa\left(\bigoplus_{\alpha\in Q_{1}}e_{\mathfrak{t}(\alpha)}A_{18}e_{\mathfrak{s}(\alpha)}\right)$.

In weight $9$, the space $\bigoplus_{\alpha\in Q_{1}}e_{\mathfrak{t}(\alpha)}A_{9}e_{\mathfrak{s}(\alpha)}$ has basis $c_{3}^{2}c_{0}c_{3}a_{0}$, $c_{3}c_{0}c_{3}^{2}a_{0}$, $(c_{0}c_{3})^{2}a_{0}$, $a_{2}^{*}c_{0}c_{2}c_{0}a_{2}a_{1}$, $c_{2}^{2}c_{0}c_{2}a_{2}$, $(c_{2}c_{0})^{2}a_{2}$, $(c_{0}c_{2})^{2}a_{2}$, $c_{0}c_{2}^{2}c_{0}a_{2}$, $a_{3}c_{3}c_{0}c_{3}^{2}$, $a_{3}(c_{0}c_{3})^{2}$, $a_{3}(c_{3}c_{0})^{2}$, $a_{3}c_{0}c_{3}^{2}c_{0}$, $a_{4}a_{3}c_{3}^{2}c_{0}a_{3}^{*}$, $a_{4}a_{3}c_{0}c_{3}^{2}a_{3}^{*}$, $a_{4}a_{3}c_{0}c_{3}c_{0}a_{3}^{*}$, $a_{5}a_{4}a_{3}c_{3}c_{0}a_{3}^{*}a_{4}^{*}$, $a_{5}a_{4}a_{3}c_{0}c_{3}a_{3}^{*}a_{4}^{*}$, $a_{6}a_{5}a_{4}a_{3}c_{0}a_{3}^{*}a_{4}^{*}a_{5}^{*}$ and $\bigoplus_{\alpha\in Q_{1}}e_{\mathfrak{s}(\alpha)}A_{9}e_{\mathfrak{t}(\alpha)}=\kappa\left(\bigoplus_{\alpha\in Q_{1}}e_{\mathfrak{t}(\alpha)}A_{9}e_{\mathfrak{s}(\alpha)}\right)$.

Finally, if $\rho_{17}$ were a coboundary, it would be equal to $b_{k}^{1}(g)$ for some $g\in\mathop{\rm Hom}\nolimits_{k^{e}}(k,A_{16})$. Such a map $g$ would necessarily satisfy $g(e_{0})\in\mathop{\rm span}\left\{a_{0}^{*}(c_{3}c_{0})^{3}c_{3}a_{0}\right\}$, $g(e_{3})\in\mathop{\rm span}\left\{(c_{3}c_{0})^{4},(c_{0}c_{3})^{4},c_{0}c_{3}(c_{3}c_{0})^{3},(c_{0}c_{3})^{3}c_{3}c_{0},c_{3}^{2}(c_{0}c_{3})^{3},c_{3}c_{0}(c_{3}c_{0}c_{3})^{2}\right\}$, and $g(e_{4})\in\mathop{\rm span}\left\{a_{3}c_{0}(c_{3}c_{0})^{3}a_{3}^{*},a_{3}c_{3}(c_{0}c_{3})^{3}a_{3}^{*},a_{3}c_{3}(c_{3}c_{0})^{3}a_{3}^{*},a_{3}(c_{0}c_{3})^{3}c_{3}a_{3}^{*}\right\}$. Then, by considering $b_{k}^{1}(g)(a_{0})$, $b_{k}^{1}(g)(a_{0}^{*})$ and $b_{k}^{1}(g)(a_{3})$ we get a contradiction.

**Proposition **\theDf

Let $A$ be a preprojective algebra of type E8.

1. \edefitit(i)

The elements in $\left\{z_{0},z_{12},z_{20},z_{24}\right\}\cup\left\{\pi_{i};i\in Q_{0}\right\}$ form a basis of $\mathop{\rm HK}\nolimits^{0}(A)$. 2. \edefitit(ii)

If $\mathop{\rm char}\nolimits(\mathbb{F})\not\in\left\{2,3,5\right\}$, the elements in $\left\{\mkern 2.0mu\overline{\mkern-2.0mu\zeta\mkern 1.0mu}\mkern-1.0mu_{\ell};\ell=0,12,20,24\right\}$ form a basis of $\mathop{\rm HK}\nolimits^{1}(A)$.

If $\mathop{\rm char}\nolimits(\mathbb{F})=2$, the elements in $\left\{\mkern 2.0mu\overline{\mkern-2.0mu\zeta\mkern 1.0mu}\mkern-1.0mu_{\ell};\ell=0,12,20,24\right\}\cup\left\{\mkern 2.0mu\overline{\mkern-2.0mu\rho\mkern 1.0mu}\mkern-1.0mu_{3},\mkern 2.0mu\overline{\mkern-2.0mu\rho\mkern 1.0mu}\mkern-1.0mu_{7},\mkern 2.0mu\overline{\mkern-2.0mu\rho\mkern 1.0mu}\mkern-1.0mu_{15},\mkern 2.0mu\overline{\mkern-2.0mu\rho\mkern 1.0mu}\mkern-1.0mu_{27}\right\}$ form a basis of $\mathop{\rm HK}\nolimits^{1}(A)$.

If $\mathop{\rm char}\nolimits(\mathbb{F})=3$, the elements in $\left\{\mkern 2.0mu\overline{\mkern-2.0mu\zeta\mkern 1.0mu}\mkern-1.0mu_{\ell};\ell=0,12,20,24\right\}\cup\left\{\mkern 2.0mu\overline{\mkern-2.0mu\rho\mkern 1.0mu}\mkern-1.0mu_{5},\mkern 2.0mu\overline{\mkern-2.0mu\rho\mkern 1.0mu}\mkern-1.0mu_{17}\right\}$ form a basis of $\mathop{\rm HK}\nolimits^{1}(A)$.

If $\mathop{\rm char}\nolimits(\mathbb{F})=5$, the elements in $\left\{\mkern 2.0mu\overline{\mkern-2.0mu\zeta\mkern 1.0mu}\mkern-1.0mu_{\ell};\ell=0,12,20,24\right\}\cup\left\{\mkern 2.0mu\overline{\mkern-2.0mu\rho\mkern 1.0mu}\mkern-1.0mu_{9}\right\}$ form a basis of $\mathop{\rm HK}\nolimits^{1}(A)$. 3. \edefitit(iii)

If $\mathop{\rm char}\nolimits(\mathbb{F})\not\in\left\{2,3,5\right\}$, the elements in $\left\{\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{j};j\in Q_{0}\right\}$ form a basis of $\mathop{\rm HK}\nolimits^{2}(A)$.

If $\mathop{\rm char}\nolimits(\mathbb{F})=2$, the elements in $\left\{\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{j};j\in Q_{0}\right\}\cup\left\{\mkern 2.0mu\overline{\mkern-2.0mu\gamma\mkern 1.0mu}\mkern-1.0mu_{4},\mkern 2.0mu\overline{\mkern-2.0mu\gamma\mkern 1.0mu}\mkern-1.0mu_{8},\mkern 2.0mu\overline{\mkern-2.0mu\gamma\mkern 1.0mu}\mkern-1.0mu_{16},\mkern 2.0mu\overline{\mkern-2.0mu\gamma\mkern 1.0mu}\mkern-1.0mu_{28}\right\}$ form a basis of $\mathop{\rm HK}\nolimits^{2}(A)$.

If $\mathop{\rm char}\nolimits(\mathbb{F})=3$, the elements in $\left\{\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{j};j\in Q_{0}\right\}\cup\left\{\mkern 2.0mu\overline{\mkern-2.0mu\gamma\mkern 1.0mu}\mkern-1.0mu_{6},\mkern 2.0mu\overline{\mkern-2.0mu\gamma\mkern 1.0mu}\mkern-1.0mu_{18}\right\}$ form a basis of $\mathop{\rm HK}\nolimits^{2}(A)$.

If $\mathop{\rm char}\nolimits(\mathbb{F})=5$, the elements in $\left\{\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{j};j\in Q_{0}\right\}\cup\left\{\mkern 2.0mu\overline{\mkern-2.0mu\gamma\mkern 1.0mu}\mkern-1.0mu_{10}\right\}$ form a basis of $\mathop{\rm HK}\nolimits^{2}(A)$.

Proof 6.13**.**

The centre was given in [22], so we have (i).

For $\mathop{\rm HK}\nolimits^{1}(A)$ and $\mathop{\rm HK}\nolimits^{2}(A)$, the number of elements in the statement is equal to the dimension of the corresponding cohomology space. Moreover, all the elements in the statement are indeed cocycles.

If $\mathop{\rm char}\nolimits(\mathbb{F})$ is not $2$, $3$ or $5$, a basis of $\mathop{\rm HK}\nolimits^{1}(A)=\mathop{\rm HH}\nolimits^{1}(A)$ was given in [22]. It consists of the classes of the $\zeta^{\prime}_{\ell}$ with $\ell\in\left\{0,12,20,24\right\}$ where $\zeta^{\prime}_{\ell}(a_{i})=a_{i}z_{\ell}$ for $0\leqslant i\leqslant 2$ and $\zeta^{\prime}_{\ell}(a_{i}^{*})=a_{i}^{*}z_{\ell}$ for $3\leqslant i\leqslant 6$. Since $\zeta_{0}-\zeta_{0}^{\prime}$ is equal to $b_{K}^{1}(e_{3}+2e_{5}+3e_{6}+4e_{7})$, and $\zeta_{\ell}-\zeta_{\ell}^{\prime}=(\zeta_{0}-\zeta_{0}^{\prime})\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}z_{\ell}$ is also a coboundary, $\zeta_{\ell}$ and $\zeta_{\ell}^{\prime}$ represent the same cohomology class for $\ell\in\left\{0,12,20,24\right\}$. Moreover, as in types A, D, E6 and E7, the elements $\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{j}$ form a basis of $\mathop{\rm HK}\nolimits^{2}(A)$.

The rest of the proof is the same as that of Proposition 6.4.1, based on Lemma 6.5.1 and the fact that the following cup products at the level of cochains are all coboundaries: $\zeta_{0}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}\rho_{27}-\gamma_{28}$, $\zeta_{12}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}\rho_{15}-\gamma_{28}$, $\zeta_{24}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}\rho_{3}-\gamma_{28}$, $\zeta_{20}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}\rho_{7}-\gamma_{28}$, $z_{24}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}\gamma_{4}-\gamma_{28}$, $z_{12}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}\gamma_{16}-\gamma_{28}$, $z_{20}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}\gamma_{8}-\gamma_{28}$, $z_{12}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}\rho_{5}-\rho_{17}$, $z_{12}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}\gamma_{6}-\gamma_{18}$, $\zeta_{0}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}\rho_{9}-\gamma_{10}$, whereas $\gamma_{28}$, $\gamma_{18}$, $\gamma_{10}$ and $\rho_{17}$ are not.

Koszul homology follows using duality (Theorem 4.3), as in Corollary 6.1.1.

6.5.2 Cup and cap products

We now determine the cup products of the elements in the bases of the Koszul cohomology spaces given above.

**Proposition **\theDf

Let $A$ be a preprojective algebra of type E8. Up to graded commutativity, the non zero cup products of elements in $\mathop{\rm HK}\nolimits^{\bullet}(A)$ are:

[TABLE]

Proof 6.14**.**

Most of the cup-products are easy to compute, follow from Lemma 6.5.1 or vanish for weight reasons. The remaining ones are obtained as follows (at the level of cochains): $z_{24}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}\rho_{3}=\rho_{27}+b_{K}^{1}([e_{3}\mapsto c_{0}(c_{3}c_{0})^{6}])$, we have $z_{12}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}\rho_{3}=\rho_{15}+b_{K}^{1}(h)$ where $h$ is defined by $h(a_{2})=(c_{0}c_{2})^{3}c_{0}a_{2}$, $h(a_{3})=a_{3}c_{0}(c_{3}c_{0})^{3}+a_{3}c_{3}(c_{0}c_{3})^{3}$, $h(a_{4})=a_{4}a_{3}(c_{0}c_{3}^{2})^{2}a_{3}^{*}$ and $h(a_{2}^{*})=a_{2}^{*}c_{0}(c_{2}c_{0})^{3}+a_{2}^{*}c_{2}(c_{0}c_{2})^{3}$, and finally $z_{12}\stackrel{{\scriptstyle[}}{{K}}]{}{\smile}\rho_{9}+\zeta_{20}=b_{K}(h^{\prime})$ where $h^{\prime}$ is defined by $h^{\prime}(e_{0})=a_{0}^{*}(c_{3}c_{0})^{4}c_{3}a_{0}$, $h^{\prime}(e_{2})=2a_{2}^{*}(c_{2}c_{0})^{4}c_{2}a_{2}$, $h^{\prime}(e_{3})=-2(c_{0}c_{3})^{5}+2c_{3}^{2}(c_{0}c_{3})^{4}+(c_{3}c_{0})^{4}c_{3}^{2}$, $h^{\prime}(e_{4})=-a_{3}(c_{0}c_{3})^{4}c_{0}a_{3}^{*}+a_{3}c_{3}(c_{3}c_{0})^{4}a_{3}^{*}-a_{3}(c_{0}c_{3})^{4}c_{3}a_{3}^{*}$ and $h^{\prime}(e_{5})=-a_{4}a_{3}(c_{0}c_{3})^{4}a_{3}^{*}a_{4}^{*}$.

The cap products follow using duality, as in Corollary 6.1.2.

6.5.3 Higher Koszul cohomology and homology

As in types A, D, E6 and E7, the differential $\partial_{\smile}^{1}$ sends $z_{\ell}$ to $2\zeta_{\ell}$ for $\ell\in\left\{0,8,12\right\}$ and the differential $\partial_{\smile}^{2}$ is zero except when $\mathop{\rm char}\nolimits(\mathbb{F})=5$ where $\partial_{\smile}^{2}(\mkern 2.0mu\overline{\mkern-2.0mu\rho\mkern 1.0mu}\mkern-1.0mu_{9})=2\mkern 2.0mu\overline{\mkern-2.0mu\gamma\mkern 1.0mu}\mkern-1.0mu_{10}$.

We then have the following higher Koszul cohomology.

**Proposition **\theDf

Let $A$ be a preprojective algebra of type E8.

If $\mathop{\rm char}\nolimits(\mathbb{F})=2$, then $\mathop{\rm HK}\nolimits^{\bullet}_{hi}(A)=\mathop{\rm HK}\nolimits^{\bullet}(A).$

If $\mathop{\rm char}\nolimits(\mathbb{F})=3$, then

[TABLE]

If $\mathop{\rm char}\nolimits(\mathbb{F})\not\in\left\{2,3\right\}$, then

[TABLE]

Higher Koszul homology follows from Theorem 4.3.

6.6 Comparison of Koszul and Hochschild (co)homology for preprojective algebras of type ADE

Let $A$ be a preprojective algebra over a Dynkin graph of type ADE. Schofield constructed a minimal projective resolution $(P^{\bullet},\partial^{\bullet})$ of $A$ as a bimodule over itself, that is periodic (of period at most $6$), which was described in [20, 22]. Following Proposition 2.8, the embedding $H(\iota^{*})_{2}$ sends the Hochschild cohomology class of an element in $\mathop{\rm Ker}\nolimits(\partial^{3}\circ-)$ to its Koszul cohomology class, and the surjection $H(\tilde{\iota})_{2}$ induces an isomorphism between $\mathop{\rm HH}\nolimits_{2}(A)$ and $\mathop{\rm HK}\nolimits_{2}(A)/\mathop{\rm Im}\nolimits(\mathop{\rm id}\nolimits_{A}\otimes\partial^{3})$.

We first transport the maps $\partial^{3}\circ-$ and $\mathop{\rm id}\nolimits_{A}\otimes\partial^{3}$ via the natural isomorphisms $A\otimes_{A^{e}}(Ae_{j}\otimes e_{i}A)\rightarrow e_{i}Ae_{j}$ that sends $\lambda\otimes(a\otimes b)$ to $b\lambda a$ and $\mathop{\rm Hom}\nolimits_{A^{e}}(Ae_{i}\otimes e_{j}A,A)\cong e_{i}Ae_{j}$ that sends $f$ to $f(e_{i}\otimes e_{j})$.

The associative non degenerate bilinear form on the selfinjective preprojective $A$ can be defined as follows, see [20] and [46, Proposition 3.15]: let $B$ be a basis of $A$ consisting of homogeneous elements, that contains the idempotents $e_{i}$, $i\in Q_{0}$ and a basis $\left\{\pi_{i}\,;\,i\in Q_{0}\right\}$ of the socle of $A$, and such that each $v\in B$ belongs to $e_{j}Ae_{i}$ for some $i,j$ in $Q_{0}.$ Then if $x\in Ae_{i}$, $(y,x)$ is the coefficient of $\pi_{i}$ in the expression of $yx$ as a linear combination of elements in $B.$ The Nakayama automorphism $\nu$ of $A$ satisfies $(y,x)=(\nu(x),y)$ for all $x,y$ in $A$, and induces a permutation of the indices, the Nakayama permutation ${\mkern 2.0mu\overline{\mkern-2.0mu\nu\mkern 1.0mu}\mkern-1.0mu}$, that is, a permutation of $Q_{0}$ such that $\mathop{\rm top}\nolimits(Ae_{i})\cong\mathop{\rm soc}\nolimits(Ae_{{\mkern 2.0mu\overline{\mkern-2.0mu\nu\mkern 1.0mu}\mkern-1.0mu}(i)})$, characterised by $\nu(e_{i})=e_{{\mkern 2.0mu\overline{\mkern-2.0mu\nu\mkern 1.0mu}\mkern-1.0mu}(i)}$.

Let $\widehat{B}$ be the dual basis of $B$ with respect to the non degenerate form $(-,-)$, so that $(\widehat{w},v)=\delta_{vw}$ for all $v,w$ in $B$. In particular, if $v\in Be_{i}$, the coefficient of $\pi_{i}$ in $\widehat{v}v$ is $1.$ Note that $v\in e_{j}Be_{i}$ if and only if $\widehat{v}\in e_{{\mkern 2.0mu\overline{\mkern-2.0mu\nu\mkern 1.0mu}\mkern-1.0mu}(i)}\widehat{B}e_{j}.$

Then the maps $\partial^{3}\circ-$ and $\mathop{\rm id}\nolimits_{A}\otimes_{A^{e}}\partial^{3}$ become respectively

[TABLE]

It then follows as in [24, Proposition 3.2.25] that $\mathop{\rm Im}\nolimits\delta_{3}$ is the span of the $\delta_{3}(e_{i})$ such that ${\mkern 2.0mu\overline{\mkern-2.0mu\nu\mkern 1.0mu}\mkern-1.0mu}(i)=i$, and that for such an $i$ we have $\delta_{3}(e_{i})=\sum_{\begin{subarray}{c}j\end{subarray}\in Q_{0}\\ {\mkern 2.0mu\overline{\mkern-2.0mu\nu\mkern 1.0mu}\mkern-1.0mu}(j)=j}\mathop{\rm tr}\nolimits(\nu_{|e_{j}Ae_{i}})\pi_{j}$. Moreover, the matrix whose coefficients are the $\mathop{\rm tr}\nolimits(\nu_{|e_{j}Ae_{i}})$ is either easy to compute or given in [22] for types D and E7. It is also known from [24] that the set of elements of weight [math] in $\mathop{\rm Ker}\nolimits\delta^{3}$ identifies with the kernel of the Cartan matrix of $A$. Moreover, for any element of positive weight $a\in A$, we have $\delta^{3}(a)=0$. Therefore $\mathop{\rm HH}\nolimits^{2}(A)$ is obtained by taking all the elements of positive weight in $\mathop{\rm HK}\nolimits^{2}(A)$ and adding the kernel of the Cartan matrix.

We shall use this as well as the dimensions of the Hochschild and Koszul (co)homology spaces to compare $\mathop{\rm HH}\nolimits^{2}(A)$ with $\mathop{\rm HK}\nolimits^{2}(A)$ and $\mathop{\rm HH}\nolimits_{2}(A)$ with $\mathop{\rm HK}\nolimits_{2}(A)$ in each case. In particular, the injection $\mathop{\rm HH}\nolimits^{2}(A)\rightarrow\mathop{\rm HK}\nolimits^{2}(A)$ is not surjective except in type E8 with $\mathop{\rm char}\nolimits(\mathbb{F})=2$.

6.6.1 Comparison of the second Koszul and Hochschild cohomology groups

In type A, the space $\mathop{\rm HH}\nolimits^{2}(A)$ was completely described by Erdmann and Snashall in [18], and they proved that $\dim\mathop{\rm HH}\nolimits^{2}(A)=n-{m_{A}}-1$ and gave a basis $\left\{\tilde{h}_{i};0\leqslant i\leqslant n-{m_{A}}-2\right\}$ with $\tilde{h}_{i}=\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{i}+\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{n-1-i}.$ The morphism of complexes $\iota^{*}_{2}$ sends $\tilde{h}_{i}$ to $h_{i}+h_{n-1-i}$, and this describes the injection $\mathop{\rm HH}\nolimits^{2}(A)\rightarrow\mathop{\rm HK}\nolimits^{2}(A)$.

In type D, if $\mathop{\rm char}\nolimits(\mathbb{F})\neq 2$ and $n$ is even, there is nothing to do since $\mathop{\rm HH}\nolimits^{2}(A)=0.$ If $\mathop{\rm char}\nolimits(\mathbb{F})\neq 2$ and $n$ is odd, then $\dim\mathop{\rm HH}\nolimits^{2}(A)=1$, the basis given in [22] for $\mathop{\rm HH}\nolimits^{2}(\Lambda_{\mathbb{C}})$ also gives a basis of $\mathop{\rm HH}\nolimits^{2}(A)$, and it is the cohomology class of the map $\psi_{0}$ defined by $\psi_{0}(\sigma_{0})=e_{0}$ and $\psi_{0}(\sigma_{1})=-e_{1}$. The embedding $\mathop{\rm HH}\nolimits^{2}(A)\rightarrow\mathop{\rm HK}\nolimits^{2}(A)$ is therefore given by $\psi_{0}\mapsto h_{0}-h_{1}.$

Now assume that $\mathop{\rm char}\nolimits(\mathbb{F})=2$. Then $\dim\mathop{\rm HH}\nolimits^{2}(A)=n+{m_{D}}-2.$ As we explained above, a basis of $\mathop{\rm HH}\nolimits^{2}(A)$ may be obtained from a basis of the set of elements of positive coefficient weight in $\mathop{\rm HK}\nolimits^{2}(A)$ to which we add elements obtained by determining a basis of the kernel of the Cartan matrix of $A$. It follows that a basis of $\mathop{\rm HH}\nolimits^{2}(A)$ is given by

[TABLE]

where $\psi_{2p+1}(\sigma_{2p+1})=e_{2p+1}$, $\psi_{2p+2}(\sigma_{2p+2})=e_{2p+2}$ and $\psi_{2p+2}(\sigma_{2})=e_{2}$, $\varphi_{2p}(\sigma_{2p})=e_{2p}$, $\varphi_{2p+1}(\sigma_{2p+3})=e_{2p+3}$ and $\varphi_{2p+1}(\sigma_{3})=e_{3}$, for $p\geqslant 1.$ Therefore, the embedding $\mathop{\rm HH}\nolimits^{2}(A)\rightarrow\mathop{\rm HK}\nolimits^{2}(A)$ fixes the $\mkern 2.0mu\overline{\mkern-2.0mu\gamma\mkern 1.0mu}\mkern-1.0mu_{\ell}$ and sends $\mkern 2.0mu\overline{\mkern-2.0mu\varphi\mkern 1.0mu}\mkern-1.0mu_{2p}$ to $\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{2p}$ and $\mkern 2.0mu\overline{\mkern-2.0mu\varphi\mkern 1.0mu}\mkern-1.0mu_{2p+3}$ to $\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{3}+\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{2p+3}$ when $n$ is even, and $\mkern 2.0mu\overline{\mkern-2.0mu\psi\mkern 1.0mu}\mkern-1.0mu_{0}$ to $\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{0}+\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{1}$, $\mkern 2.0mu\overline{\mkern-2.0mu\psi\mkern 1.0mu}\mkern-1.0mu_{2p+1}$ to $\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{2p+1}$ and $\mkern 2.0mu\overline{\mkern-2.0mu\psi\mkern 1.0mu}\mkern-1.0mu_{2p+2}$ to $\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{2}+\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{2p+2}$ when $n$ is odd.

In type E6, the Cartan matrix is equivalent, through row operations, to

[TABLE]

Let $\varphi_{0}$, $\varphi_{1}$, $\varphi_{2}$ and $\varphi_{3}$ be the maps in $\mathop{\rm Hom}\nolimits_{k^{e}}(R,A)$ defined by $\varphi_{0}(\sigma_{0})=e_{0}$, $\varphi_{1}(\sigma_{1})=e_{1}$, $\varphi_{1}(\sigma_{5})=-e_{5}$, $\varphi_{2}(\sigma_{2})=e_{2}$, $\varphi_{2}(\sigma_{4})=-e_{4}$ and $\varphi_{3}(\sigma_{3})=e_{3}$. Then $\mathop{\rm HH}\nolimits^{2}(A)$ is the subspace of $\mathop{\rm HK}\nolimits^{2}(A)$ spanned by

[TABLE]

In type E7, if $\mathop{\rm char}\nolimits(\mathbb{F})\neq 2$, we have $\dim\mathop{\rm HH}\nolimits^{2}(A)=\dim\mathop{\rm HK}\nolimits^{2}(A)_{>0}$, so that $\mathop{\rm HH}\nolimits^{2}(A)$ is precisely the subspace of $\mathop{\rm HK}\nolimits^{2}(A)$ of elements of positive weight (which is zero unless $\mathop{\rm char}\nolimits(\mathbb{F})\in\left\{2,3\right\})$. If $\mathop{\rm char}\nolimits(\mathbb{F})=2$, then the Cartan matrix of $A$ is equivalent, through row operations, to the matrix $\left(\begin{smallmatrix}1&0&0&0&1&0&1\end{smallmatrix}\right)$ so that $\mathop{\rm HH}\nolimits^{2}(A)$ is the subspace of $\mathop{\rm HK}\nolimits^{2}(A)$ spanned by $\mkern 2.0mu\overline{\mkern-2.0mu\gamma\mkern 1.0mu}\mkern-1.0mu_{4}$, $\mkern 2.0mu\overline{\mkern-2.0mu\gamma\mkern 1.0mu}\mkern-1.0mu_{8}$, $\mkern 2.0mu\overline{\mkern-2.0mu\gamma\mkern 1.0mu}\mkern-1.0mu_{16}$, $\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{1}$, $\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{2}$, $\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{3}$, $\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{5}$, $\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{0}+\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{4}$ and $\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{0}+\mkern 2.0mu\overline{\mkern-2.0muh\mkern 1.0mu}\mkern-1.0mu_{6}$.

In type E8, we only need to look at dimensions. Indeed, if $\mathop{\rm char}\nolimits(\mathbb{F})=2$, then $\dim\mathop{\rm HH}\nolimits^{2}(A)=\dim\mathop{\rm HK}\nolimits^{2}(A)$ so that $\mathop{\rm HH}\nolimits^{2}(A)\cong\mathop{\rm HK}\nolimits^{2}(A)$, and if $\mathop{\rm char}\nolimits(\mathbb{F})\neq 2$ then $\dim\mathop{\rm HH}\nolimits^{2}(A)=\dim\mathop{\rm HK}\nolimits^{2}(A)_{>0}$ so that $\mathop{\rm HH}\nolimits^{2}(A)$ is precisely the subspace of $\mathop{\rm HK}\nolimits^{2}(A)$ of elements of positive weight (which is zero unless $\mathop{\rm char}\nolimits(\mathbb{F})\in\left\{2,3,5\right\})$.

6.6.2 Comparison of the second Koszul and Hochschild homology groups

In type A, if $n$ is even or if $n$ is odd and $\mathop{\rm char}\nolimits(\mathbb{F})$ divides $({m_{A}}+1)$, we have seen that $\dim\mathop{\rm HK}\nolimits_{2}(A)=\dim\mathop{\rm HH}\nolimits_{2}(A)$ and therefore $\mathop{\rm HK}\nolimits_{2}(A)\cong\mathop{\rm HH}\nolimits_{2}(A)$. Now assume that $n$ is odd and $\mathop{\rm char}\nolimits(\mathbb{F})\nmid({m_{A}}+1)$. Then $\dim\mathop{\rm HK}\nolimits_{2}(A)=\dim\mathop{\rm HH}\nolimits_{2}(A)+1$. We must determine the image of the map $\delta_{3}$ given in Subsection 6.6.

The Nakayama permutation ${\mkern 2.0mu\overline{\mkern-2.0mu\nu\mkern 1.0mu}\mkern-1.0mu}$ has precisely one fixed point, which is ${m_{A}}$. The matrix of $\nu$ restricted to $e_{m_{A}}Ae_{m_{A}}$ is the identity matrix $I_{{m_{A}}+1}$. Consequently, $\delta_{3}(e_{{m_{A}}})=({m_{A}}+1)\pi_{m_{A}}$ spans the image of $\delta_{3}.$ Via the isomorphism $\bigoplus_{i\in Q_{0}}e_{i}Ae_{i}\cong A\otimes_{k^{e}}R$, $\pi_{m_{A}}$ corresponds to $\check{z}_{m_{A}}$ so that we have $\mathop{\rm HH}\nolimits_{2}(A)\cong\mathop{\rm HK}\nolimits_{2}(A)/\mathop{\rm span}\left\{\check{z}_{m_{A}}\right\}$.

In type D, if $n$ is odd and $\mathop{\rm char}\nolimits(\mathbb{F})=2$, then we know that $\dim\mathop{\rm HK}\nolimits_{2}(A)=\dim\mathop{\rm HH}\nolimits_{2}(A)$ and therefore $\mathop{\rm HK}\nolimits_{2}(A)\cong\mathop{\rm HH}\nolimits_{2}(A)$. In the other cases, we need to determine the image of the map $\delta_{3}$. We should note that the Nakayama automorphism in [22] differs from $\nu$ by composition with the inner automorphism given by the invertible element $u=-e_{0}+\sum_{i=1}^{n-1}(-1)^{i}e_{i}$ (after changing the labelling of vertices and arrows so that they are the same as ours).

If $n$ is odd and $\mathop{\rm char}\nolimits(\mathbb{F})\neq 2$, the fixed points of the Nakayama permutation are the integers $i$ with $2\leqslant i\leqslant n-1.$ The matrix $(n-2)\times(n-2)$ matrix $H^{\nu}$ whose $(i,j)$-coefficient is the trace of $\nu$ restricted to $e_{j}Ae_{i}$ was given in [22, paragraph 11.2.3]; we need only change the signs of coefficients $(i,j)$ with $i$ or $j$ (but not both) equal to [math] or odd, so that $\mathop{\rm tr}\nolimits\left(\nu_{|e_{j}Ae_{i}}\right)=2$ if $i$ and $j$ are even and is equal to [math] otherwise. Therefore for all $i$ fixed by ${\mkern 2.0mu\overline{\mkern-2.0mu\nu\mkern 1.0mu}\mkern-1.0mu}$, we have $\delta_{3}(e_{i})=\sum_{p=1}^{{m_{D}}+1}2\pi_{2p}.$ Using the isomorphism $\bigoplus_{i\in Q_{0}}e_{i}Ae_{i}\cong A\otimes_{k^{e}}R$, we obtain $\mathop{\rm HH}\nolimits_{2}(A)\cong\mathop{\rm HK}\nolimits_{2}(A)/\mathop{\rm span}\left\{\sum_{p=1}^{{m_{D}}+1}\check{\pi}_{2p}\right\}$.

If $n$ is even, all the integers $i$ with $0\leqslant i\leqslant n-1$ are fixed points of ${\mkern 2.0mu\overline{\mkern-2.0mu\nu\mkern 1.0mu}\mkern-1.0mu}.$ The matrix $H^{\nu}$ was given in [22, paragraph 11.2.2] and the same adaptations as in the case $n$ odd gives the $n\times n$ symmetric matrix

[TABLE]

It follows that, if $n$ is even,

[TABLE]

In types E6 and E8, since they have the same dimensions, the homology spaces $\mathop{\rm HH}\nolimits_{2}(A)$ and $\mathop{\rm HK}\nolimits_{2}(A)$ are isomorphic.

Finally, in type E7, if $\mathop{\rm char}\nolimits(\mathbb{F})=3$, then $\mathop{\rm HH}\nolimits_{2}(A)\cong\mathop{\rm HK}\nolimits_{2}(A)$. Now assume that $\mathop{\rm char}\nolimits(\mathbb{F})\neq 3$. The matrix $H^{\nu}$ was given in [22]; here again, the Nakayama automorphism in [22] differs from $\nu$ by composing with the inner automorphism associated with the element $-e_{0}+e_{1}-e_{2}+e_{3}+e_{4}+e_{5}+e_{6}$ (after change of labelling) and therefore the non zero rows of the matrix $H^{\nu}$ are those corresponding to vertices $0,$ $4$ and $6$ and are all equal to $\left(\begin{smallmatrix}3&0&0&0&3&0&3\end{smallmatrix}\right).$ It follows that $\mathop{\rm HH}\nolimits_{2}(A)\cong\mathop{\rm HK}\nolimits_{2}(A)/\mathop{\rm span}\left\{\check{\pi}_{0}+\check{\pi}_{4}+\check{\pi}_{6}\right\}$.

6.7 A minimal complete list of cohomological invariants

**Theorem **\theDf

Let $A$ be the preprojective algebra of a Dynkin graph $\Delta$ over $\mathbb{F}$. Assume that $A$ has type either $\mathrm{A}_{n}$ with $n\geqslant 3$, or $\mathrm{D}_{n}$ with $n\geqslant 4$, or $\mathrm{E}_{n}$ with $n=6,\,7,\,8$. Let $A^{\prime}$ be a preprojective algebra of type $\mathrm{ADE}$, where the integer $n^{\prime}$ concerning $A^{\prime}$ is subjected to the same assumptions. Denote by $(d_{p})$ the equality $\dim\mathop{\rm HK}\nolimits^{p}_{hi}(A)=\dim\mathop{\rm HK}\nolimits^{p}_{hi}(A^{\prime})$. If $(d_{p})$ holds for $p=0,1$ and $2$, then $n=n^{\prime}$, and $A$ and $A^{\prime}$ have the same type. The conclusion of this implication does not hold if $(d_{2})$ is removed from the assumption.

Proof 6.15**.**

We apply the results contained in Propositions 6.1.3, 6.2.3, 6.3.3, 6.4.3 and 6.5.3. The implication is a consequence of the following items.

1. (1)

Assume that $A$ and $A^{\prime}$ have types A or D. If $\mathop{\rm char}\nolimits\mathbb{F}\neq 2$, then $n=n^{\prime}$ by $(d_{2})$, and $A$ and $A^{\prime}$ have the same type by $(d_{0})$. If $\mathop{\rm char}\nolimits\mathbb{F}=2$ and $A$ and if $A^{\prime}$ both have type A, then $n=n^{\prime}$ by $(d_{2})$. If $\mathop{\rm char}\nolimits\mathbb{F}=2$, $A$ is of type $A_{n}$ and $A^{\prime}$ is of type $D_{n^{\prime}}$, then the sum of $(d_{0})$ and $(d_{1})$ shows that $n=2n^{\prime}+m^{\prime}_{D}-2$ or $-3$ which contradicts $(d_{2})$: $n=n^{\prime}+m^{\prime}_{D}$. If $\mathop{\rm char}\nolimits\mathbb{F}=2$ and if $A$ and $A^{\prime}$ both have type D, then $n=n^{\prime}$ by $(d_{1})$. 2. (2)

If $A$ and $A^{\prime}$ have type E, then $n=n^{\prime}$ by $(d_{0})$. 3. (3)

If $A$ is of type $A_{n}$ and $A^{\prime}$ of type $E_{6}$, then $(d_{2})$ implies either that $n=6$ if $\mathop{\rm char}\nolimits\mathbb{F}\neq 2,3$ or that $n=7$ if $\mathop{\rm char}\nolimits\mathbb{F}=2,3$, but each case is excluded by $(d_{0})$. Similarly when $A^{\prime}$ is of type $E_{7}$ and $E_{8}$. 4. (4)

If $A$ is of type $D_{n}$ and $A^{\prime}$ is of type $E_{6}$, then $(d_{2})$ implies one of the three following cases: $n=6$ if $\mathop{\rm char}\nolimits\mathbb{F}\neq 2,3$, $n=7$ if $\mathop{\rm char}\nolimits\mathbb{F}=3$ or $n+m_{D}=7$ if $\mathop{\rm char}\nolimits\mathbb{F}=2$, but each case is excluded by $(d_{0})$. Similarly when $A^{\prime}$ is of type $E_{7}$ and $E_{8}$ (we also use $(d_{1})$ for $E_{8}$).

Let us show that we cannot remove assumption $(d_{2})$. It is clear if $\mathop{\rm char}\nolimits\mathbb{F}\neq 2$ because when $A$ is of type $A_{3}$ and $A^{\prime}$ is of type $A_{5}$, both $(d_{0})$ and $(d_{1})$ hold. If $\mathop{\rm char}\nolimits\mathbb{F}=2$, we check that when $A$ is of type $A_{9}$ and $A^{\prime}$ is of type $E_{6}$ then $(d_{0})$ and $(d_{1})$ hold.

It is obvious that $\mathop{\rm HK}\nolimits^{\bullet}_{hi}(A)\cong\mathbb{F}$ if $A$ is of type $A_{1}$ and it is easy to check that $\mathop{\rm HK}\nolimits^{\bullet}_{hi}(A)=0$ when $A$ is of type $A_{2}$, therefore we have obtained a minimal complete list of cohomological invariants for all the ADE preprojective algebras.

Another direct application of our computations is the following. If $A$ is as in Theorem 6.7 and if $\mathop{\rm char}\nolimits\mathbb{F}\neq 2$, the product of the algebra $\mathop{\rm HK}\nolimits^{\bullet}_{hi}(A)$ is identically zero. If $\mathop{\rm char}\nolimits\mathbb{F}=2$, $\mathop{\rm HK}\nolimits^{\bullet}_{hi}(A)=\mathop{\rm HK}\nolimits^{\bullet}(A)$ is a unital algebra whose product is fully described in our results. {Rm} In the one vertex case, the higher Koszul homology and cohomology play an essential role in a specific formulation of a Koszul Poincaré Lemma (a Koszul Poincaré duality), see Conjectures 6.5 and 7.2 in [7]. For this reason, we have formulated Theorem 6.7 in terms of the higher Koszul cohomology. An analogous statement with Koszul cohomology is also true and follows in the same way from our results, but in this case the minimality depends on the characteristic.