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
01
Koszul calculus vanishes in degrees p>2 for certain preprojective algebras
02
Isomorphism between (co)homological calculus via degree exchange p and 2-p
03
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.
HK∙(A,M)=H∙(M⊗AeK(A)) and HK∙(A,M)=H∙(HomAe(K(A),M)).
HK∙(A,M)=H∙(M⊗AeK(A)) and HK∙(A,M)=H∙(HomAe(K(A),M)).
χ~=M⊗Aeχ:M⊗AeK(A)→M⊗AeB(A),
χ~=M⊗Aeχ:M⊗AeK(A)→M⊗AeB(A),
χ∗=HomAe(χ,M):HomAe(B(A),M)→HomAe(K(A),M).
χ∗=HomAe(χ,M):HomAe(B(A),M)→HomAe(K(A),M).
HK∙(A,M)
HK∙(A,M)
HK∙(A,M)
bpK(m⊗kex1…xp)
bpK(m⊗kex1…xp)
bKp+1(f)(x1…xp+1)
HKp(A,k)≅k⊗keWp≅i∈Q0⨁eiWpei,
HKp(A,k)≅k⊗keWp≅i∈Q0⨁eiWpei,
HKp(A,k)≅Homke(Wp,k)≅i∈Q0⨁Hom(eiWpei,F).
HKp(A,k)≅Homke(Wp,k)≅i∈Q0⨁Hom(eiWpei,F).
(fK⌣g
(fK⌣g
fK⌢z
zK⌢f
(fK⌣g)K⌣h
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We show that the Koszul calculus of a preprojective algebra, whose graph is distinct from A1 and
A2, 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 Poincaré Van den Bergh duality theorem. We compute explicitly the Koszul calculus of preprojective algebras for the ADE Dynkin graphs.
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.
Erdmann and Snashall [18, 19] determined the Hochschild cohomology and its
cup-product in type A.
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]).
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].
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].
Eu and Schedler extended the ADE results to the case where the base ring is 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 Δ distinct from A1 and A2, over a field F. Let M be an A-bimodule.
\edefitit(i)
The Koszul bimodule complex K(A) of A has length 2. In particular,
HKp(A,M)=HKp(A,M)=0 for any p>2.
2. \edefitit(ii)
The HK∙(A)-bimodules HK∙(A,M) and HK2−∙(A,M) are
isomorphic.
3. \edefitit(iii)
The HKhi∙(A)-bimodules HKhi∙(A,M) and
HK2−∙hi(A,M) are isomorphic.
In this statement, following [7], HKp(A,M) and HKp(A,M) denote the Koszul homology and cohomology spaces with coefficients in M, while HKphi(A,M) and HKhip(A,M) denote the higher Koszul homology and cohomology spaces. When M=A, these notations are simplified into HKp(A), HKp(A), HKphi(A) and HKhip(A).
In the general setting [7], the Koszul calculus of a quadratic algebra A consists of the graded associative algebra HK∙(A) endowed with the Koszul cup product and, for all A-bimodules M, of the graded HK∙(A)-bimodules HK∙(A,M) and HK∙(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 HK∙(A) and of the graded HK∙(A)-bimodule HK∙(A) – similarly for the higher version.
Using Theorem 1 for Δ 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 C1 of Koszul cochains with coefficients in M, whose pth cohomology is HKp(A,M), to the complex C2 of Koszul chains with coefficients in M, whose pth homology is HK2−p(A,M), described as follows.
**Proposition **\theDf
Let A be the preprojective algebra of a connected graph Δ distinct from A1 and A2. Let M be an A-bimodule. The Koszul cup and cap products are denoted by K⌣ and K⌢. Define ω0=∑iei⊗σi, where the sum runs over the vertices i of Δ and, for each vertex i, ei is the idempotent and σi 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 θM(f) with coefficients in M by
[TABLE]
Then θM:C1→C2 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 θM with fundamental formulas of Koszul calculus [7], using actions involving K⌣ and K⌢. The fundamental formulas of Koszul calculus express the differential bK of C1 and the differential bK of C2 respectively as a cup bracket and a cap bracket, namely
[TABLE]
where eA:V→A is a fundamental Koszul 1-cocycle defined on the arrow space V by eA(x)=x for all x∈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 Ae-module M=Ae:=A⊗Aop viewed as an A-bimodule. We show that the complex of Koszul chains with coefficients in the left Ae-module Ae is naturally isomorphic, as a right Ae-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 Δ distinct from A1 and A2, over a field F. Let us denote by K(A) the Koszul bimodule complex of A. Then the A-bimodule HKp(A,Ae) is isomorphic to the A-bimodule H2−p(K(A)) for 0⩽p⩽2. In particular, we have the following.
\edefitit(i)
The A-bimodule HK2(A,Ae) is isomorphic to the A-bimodule A.
2. \edefitit(ii)
HK1(A,Ae)=0.
3. \edefitit(iii)
The A-bimodule HK0(A,Ae) is isomorphic to the A-bimodule H2(K(A)), which is
always non-zero when Δ is Dynkin of type 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 F-linear isomorphism
[TABLE]
we say that the class ω0∈HK2(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 HK∙(A) on HK∙(A,Ae) and H(K(A)). These actions can be viewed as graded actions of left HK∙(A)e-modules, while the A-bimodules can be viewed as compatible right Ae-modules. So the isomorphism HK∙(A,Ae)≅H2−∙(K(A)) in Theorem 1 is an isomorphism of graded HK∙(A)e-Ae-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 Δ 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 Δ 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 HHp(A) for infinitely many values of p. Even taking into account this 6-periodicity, the list of cohomological Koszul invariants consists only of HK0(A), HK1(A) and HK2(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 Q=(Q0,Q1) be a finite quiver, and let F be a field. Let A be an F-algebra defined on the path algebra FQ of Q by homogeneous quadratic relations. Define the ring k=FQ0,
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⩾0, if
\edefnit(i)
the bimodule Koszul complex K(A) of A has length n, and
2. \edefnit(ii)
RHomAe(K(A),Ae)≅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 HKp(A,Ae) for 0⩽p⩽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 F-vector spaces HKp(A,M) and HKn−p(A,M) are isomorphic.
{Df}
Let A be a Kc-Calabi-Yau algebra of dimension n. The image c∈HKn(A) of the unit 1 of the algebra A under the isomorphism HK0(A)≅HKn(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 A~=HomAe(K(A),A). The complexes K(A) and HomAe(K(A),Ae) of A-bimodules have an enriched structure since they can be viewed as DG A~-bimodules in the abelian category A-Bimod of A-bimodules, in the sense of [45].
Denote by C(A~,A-Bimod) the category of DG A~-bimodules in A-Bimod [45]. Let M be an A-bimodule. For any chain DG A~-bimodule C in A-Bimod, HomAe(C,M) is a cochain DG A~-bimodule in the abelian category VectF of F-vector spaces (in A-Bimod when M=Ae). For any cochain DG A~-bimodule C′ in A-Bimod, M⊗AeC′ is a cochain DG A~-bimodule in VectF. The bounded derived categories Db(A~,A-Bimod) and \mathcal{D}^{b}(\tilde{A},\text{{Vect}{}{\mathbb{F}}}) are defined in [45]. However we do not know if the functors HomAe(−,M) and M⊗Ae− from Cb(A~,A-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 HomAe(−,Ae) of Cb(A~,A-Bimod) exists and if RHomAe(K(A),Ae)≅K(A)[−n] in the bounded derived category Db(A~,A-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 HomAe(−,A) and A⊗Ae− from Cb(A~,A-Bimod) to \mathcal{C}^{b}(\tilde{A},\text{{Vect}{}_{\mathbb{F}}}) exist. Then
[TABLE]
is an isomorphism of HK∙(A)-bimodules, inducing an isomorphism of HKhi∙(A)-bimodules from HKhi∙(A) to HKn−∙hi(A). For all α∈HKp(A), we have cK⌢α=(−1)npαK⌢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 HKhi0(A), HKhi1(A) and HKhi2(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 Q be a finite quiver, meaning that the vertex set Q0 and the arrow set Q1 are finite. Let F be a field. The vertex space k=FQ0 becomes a commutative ring by associating with Q0 a complete set of orthogonal idempotents {ei;i∈Q0}. The ring k is isomorphic to F∣Q0∣, where ∣Q0∣ is the cardinal of Q0. Throughout the paper, the case ∣Q0∣=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 α∈Q1, denote its source vertex by s(α) and its target vertex by t(α). The arrow space V=FQ1 is a k-bimodule for the following actions: ejαei is equal to zero if i=s(α) or j=t(α), and is equal to α if i=s(α) and j=t(α).
Via the ring morphism F→k that maps 1 to ∑i∈Q0ei, the tensor k-algebra Tk(V) of the k-bimodule V is an F-algebra isomorphic to the path algebra FQ, so that V⊗km is identified with FQm, where Qm is the set of paths of length m. For two arrows α and β, note that
α⊗kβ is zero if t(β)=s(α), and otherwise α⊗kβ is identified with the composition αβ of paths (where paths are written from right to left, as in [4]).
Let R be a sub-k-bimodule of V⊗kV≅FQ2. The unital associative
k-algebra A=Tk(V)/(R), where (R) denotes the two-sided ideal of Tk(V) generated by R, is
called a quadratic k-algebra over the finite quiver 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 Am. Clearly, A0≅k and A1≅V. The algebra A is F-central, meaning that the left action of λ∈F on A is the same as its right action. However if there is an arrow α joining two distinct vertices i and j, A is not k-central since the left and right actions of ei on α 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 F-symmetric. Setting Ae=A⊗FAop, any A-bimodule can be viewed as a left (or right) Ae-module, as usual.
For brevity, the notation ⊗F is replaced by the unadorned tensor product ⊗. Similarly for the notations HomF and dimF abbreviated to Hom and dim. If unspecified, a vector space is an F-vector space and a linear map is F-linear.
The tensor product ⊗k is different from the unadorned tensor product ⊗. However, if
M is a right A-module and N is a left A-module,
then the natural linear map
Mei⊗eiN→Mei⊗keiN
is an isomorphism, so that for a∈Mei and b∈eiN, we can identify a⊗kb=a⊗b. Similarly, if M and N are A-bimodules, ejMei and eiNej are k-bimodules, that may be viewed as left and right ke-modules, where ke=k⊗k. The natural linear map
ejMei⊗eiNej→ejMei⊗keeiNej
is an isomorphism, so for a∈ejMei and b∈eiNej, we can identify a⊗keb=a⊗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⊗kA⊗k∙⊗kA,d) with
[TABLE]
for a, a′ and a1,…,ap 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⊗kA⊗kp⊗kA 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⊗kp and transporting M⊗Aed and HomAe(d,M) via these isomorphisms, we obtain the Hochschild differentials bH and bH, so that
[TABLE]
[TABLE]
The Hochschild homology differential is then defined, for m∈M and a1,…,ap in A, by
[TABLE]
The Hochschild cohomology differential (including a Koszul sign in HomAe(d,M)) is defined, for f∈Homke(A⊗kp,M) and a1,…,ap+1 in A, by
[TABLE]
2.2 Koszul homology and cohomology
Let A=Tk(V)/(R) be a quadratic k-algebra over 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⊗kWp⊗kA of A⊗kA⊗kp⊗kA, where W0=k, W1=V and, for p⩾2,
[TABLE]
Here Wp is considered as a sub-k-bimodule of V⊗kp⊆A⊗kp. It is immediate that the differential d of K(A) is defined on A⊗kWp⊗kA by
[TABLE]
for a, a′ in A and x1…xp in Wp.
In this paper, we systematically follow [7] for the notation of elements of Wp. Let us recall this notation. As in (3), an arbitrary element of Wp is denoted by a product x1…xp thought of as a sum of such products, where x1,…,xp are in V. Moreover, regarding Wp as a subspace of V⊗kq⊗kWr⊗kV⊗ks with q+r+s=p, the
element x1…xp viewed in V⊗kq⊗kWr⊗kV⊗ks will be denoted by the same notation, meaning that
the product xq+1…xq+r represents an element of Wr and the other xi 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 μ:A⊗kA→A the multiplication of A. Then A is Koszul if
and only if μ:K(A)→A is a resolution of A. If R=0 and if R=V⊗kV, 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⊗kWp⊗kA forming K(A) are projective and finitely
generated. Indeed, Wp is a sub-k-bimodule of V⊗kp, 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.
\edefitit(i)
The A-bimodule A⊗kE⊗kA is projective.
2. \edefitit(ii)
If E is finite dimensional, then the A-bimodule A⊗kE⊗kA is finitely
generated.
Proof 2.1**.**
Clearly E=⨁i,j∈Q0ejEei. From A=⨁i∈Q0Aei=⨁j∈Q0ejA, we deduce that the A-bimodule A⊗kE⊗kA is isomorphic to the A-bimodule
[TABLE]
Considering F1 as a sub-A-bimodule of F=A⊗E⊗A, we see that F=F1⊕F2, where
[TABLE]
in which the sum is taken over the set of indices with i1=i2 and i3=i4. As the A-bimodule F is free, we conclude that F1 is projective. Part (ii) follows from the fact that E is finite dimensional if and only if all the (ejEei) 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 HK∙(A)=HK∙(A,A) and HK∙(A)=HK∙(A,A).
Since K(A) is a complex of projective A-bimodules, M↦HK∙(A,M) and M↦HK∙(A,M) define δ-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], HKp(A,M)
(respectively HKp(A,M)) is isomorphic to a Hochschild hyperhomology (respectively hypercohomology) space.
The inclusion χ:K(A)→B(A) is a morphism of complexes that induces the following morphisms of complexes
[TABLE]
[TABLE]
The linear maps H(χ~):HKp(A,M)→HHp(A,M) and H(χ∗):HHp(A,M)→HKp(A,M) are always isomorphisms for p=0 and p=1, and if A is Koszul they are isomorphisms for any p.
Taking E=Wp in the isomorphisms (1), we get isomorphisms
[TABLE]
with differentials bK:M⊗keWp→M⊗keWp−1 and bK:Homke(Wp,M)→Homke(Wp+1,M) given by
[TABLE]
where m∈M and x1…xp∈Wp, respectively f∈Homke(Wp,M) and x1…xp+1∈Wp+1.
Note that the k-algebra A is augmented by the natural projection ϵA:A→A0≅k. Let us examine now the particular case M=k, where k is the A-bimodule defined by ϵA. The action on k of an element of Ap with p>0 is zero, so that the Koszul differentials vanish when M=k. Consequently, we have the linear isomorphisms
[TABLE]
[TABLE]
In particular HKp(A,k)≅Hom(HKp(A,k),F), generalising [7, Proposition 2.8].
2.3 Koszul cup and cap products
Let A=Tk(V)/(R) be a quadratic k-algebra over Q. As in [7], the usual cup and cap products ⌣ and ⌢ in Hochschild cohomology and homology provide, by restriction from B(A) to K(A), the Koszul cup and cap products K⌣ and K⌢ 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∈Homke(Wp,P), g∈Homke(Wq,Q) and z=m⊗kex1…xq∈M⊗keWq, we define fK⌣g∈Homke(Wp+q,P⊗AQ), fK⌢z∈(P⊗AM)⊗keWq−p and zK⌢f∈(M⊗AP)⊗keWq−p by
[TABLE]
For any Koszul cochains f, gh 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∈Homke(Wp,P) and
g∈Homke(Wq,Q), we have the identity
[TABLE]
so that (Homke(W∙,A),bK,K⌣) is a DG algebra. This DG algebra
will play an essential role in Section 3 and will be denoted by A~. Note that
H(A~)=HK∙(A) is a graded algebra for K⌣. Moreover, formula
(9) and analogous formulas for bK(fK[]⌢z) and bK(zK[]⌢f) show that for any A-bimodule M, Homke(W∙,M) and M⊗keW∙ are DG bimodules over A~ for the actions of K⌣ and K⌢ respectively, so that HK∙(A,M) and HK∙(A,M) are graded HK∙(A)-bimodules.
{Df}
Let A=Tk(V)/(R) be a quadratic k-algebra over a finite quiver Q.
\edefnit(i)
The general Koszul calculus of A is the datum of all the spaces HK∙(A,P) and HK∙(A,M) endowed with K⌣ and K⌢, when the A-bimodules P and M vary.
2. \edefnit(ii)
The Koszul calculus of A consists of the graded associative algebra HK∙(A) and of
all the graded HK∙(A)-bimodules HK∙(A,M) and HK∙(A,M), when the
A bimodule M vary.
3. \edefnit(iii)
The restricted Koszul calculus of A consists of the graded associative algebra
HK∙(A) and of the graded HK∙(A)-bimodule HK∙(A).
4. \edefnit(iv)
The scalar Koszul calculus of A consists of the graded associative algebra
HK∙(A,k) and of the graded HK∙(A,k)-bimodule HK∙(A,k).
Since HK0(A)=Z(A) is the centre of the algebra A, the spaces HKp(A,M) and HKp(A,M) are
symmetric Z(A)-bimodules (left and right actions coincide). However HKp(A,M) and HKp(A,M)
are not k-bimodules in general. Indeed, HK0(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 ei1=ei is not in Z(A).
{Ex}
If Q1=∅, 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 A3. 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 A3 over F=C, that is, the C-algebra defined by the quiver
[TABLE]
subject to the relations
[TABLE]
The algebra A has dimension 10 and a basis of A over C is given by the elements ei for 0⩽i⩽2, ai and ai∗ for 0⩽i⩽1, a1∗a1, a1a0 and a0∗a1∗.
We then have W0=CQ0=C⟨e1,e1,e2⟩=k, W1=CQ1=C⟨a1,a1,a0∗,a1∗⟩=V and W2=C⟨σ0,σ1,σ2⟩=R.
Now consider W3=(V⊗kR)∩(R⊗kV), viewed inside CQ3. An element u in W3 can therefore be written as a path in CQ3 in two ways:
[TABLE]
with λi, λi∗, μi, μi∗ in C. Then, in CQ3, we have
[TABLE]
so that all the coefficients λi, λi∗, μi and μi∗ must be zero, hence u=0 and W3=0.
Since Wp=(Wp−1⊗kV)∩(V⊗kWp−1) for all p⩾3, it follows that Wp=0 for all p⩾3. This is true for any preprojective algebra of type
Δ with Δ different from A1 and A2, see Theorem 4.2.
The Koszul complex K(A) is therefore
[TABLE]
with
[TABLE]
Applying HomAe(−,A) and using the natural isomorphism HomAe(A⊗kE⊗kA,A)≅Homke(E,A) for any k-bimodule E (1), we get the complex
[TABLE]
Before we describe the maps, let us note that Homke(k,A)≅⨁i=02eiAei has basis {e0,e1,e2,a1∗a1}, that a general element f in Homke(V,A) is defined by f(ai)=λiai and f(ai∗)=λi∗ai∗ for i=0,1 with λi,λi∗ in C, and that a general element g in Homke(R,A) is defined by g(σi)=αiei for i=0,2 and g(σ2)=α2e1+βa1∗a1 for some scalars αi and β. Then
[TABLE]
It is then easy to see that HK0(A)=C⟨z0=1,z1=a1∗a1⟩, that HK1(A)=C⟨ζ0⟩ with ζ0∈Homke(V,A) defined by ζ0(ai)=ai and ζ0(ai∗)=0, and that HK2(A)=C⟨h0,h1,h2⟩ with hi∈Homke(R,A) defined by hi(σj)=δijei.
Moreover, the fundamental 1-cocycle eA is equal to 2ζ0+bK1(2e0+e1).
The Koszul cup products can easily be found using the formula (6). It follows that K[]⌣ is graded commutative, that 1K[]⌣x=x for any x∈HK∙(A) and that all other cup products are [math] in HK∙(A). For instance, z1K[]⌣h1 is the coboundary bK2(f) where f sends a0 to a0 and all other arrows to [math].
In particular, eAK[]⌣eA=0.
In order to determine the Koszul homology of A,
we could also apply the functor A⊗Ae− 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 ω0=∑i=02ei⊗σi∈A⊗keR. There is an isomorphism θA:HK∙(A)→HK2−∙(A) given by f↦ω0K[]⌢f. Explicitly in our example,
θA(z0)=ω0 and θA(z1)=z1⊗σ1 form a basis of HK2(A);
θA(ζ0)=a0⊗a0∗+a1⊗a1∗ forms a basis of HK1(A);
θA(hi)=ei⊗ei for 0⩽i⩽2 form a basis of HK0(A).
The Koszul cap products can also be obtained using duality and they all vanish except the cap products z0K[]⌢x=x=xK[]⌢z0 for all x∈HK∙(A).
2.4 Fundamental formulas of Koszul calculus
Let A=Tk(V)/(R) be a quadratic k-algebra over 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∈Homke(Wp,P), g∈Homke(Wq,Q), z∈M⊗keWq. 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→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 DA of the graded algebra A. This Koszul 1-cocycle is denoted by eA and is called the fundamental 1-cocycle. Its Koszul class is denoted by eA and is called the fundamental 1-class. In the one vertex case, eA is not a coboundary if V=0 [7], but this property does not hold in general.
**Lemma **\theDf
Let A=Tk(V)/(R) be a quadratic k-algebra over Q with Q1=∅. If the underlying graph of Q is simple, that is, it contains neither loops nor multiple edges, then eA is a coboundary.
Proof 2.2**.**
The 1-cocycle eA is a coboundary if and only if there exists a ke-linear map c:k→k such that eA=bK(c). Such a map is of the form c(ei)=λiei with
λi∈F, for all i∈Q0. Then eA=bK(c) if and only if
λt(α)−λs(α)=1 for any α∈Q1. The assumption on the graph means that Q has no loop and that given two
distinct vertices, there is at most one arrow joining them. Then we can choose λt(α)=1 and λs(α)=0.
This proof shows that if the quiver Q has a loop, eA is not a coboundary. The same conclusion holds if charF=2 and 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=Tk(V)/(R) be a quadratic k-algebra over 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=Tk(V)/(R) be a quadratic k-algebra over Q and let M be an A-bimodule. For
any α∈HKp(A,M) with p=0 or p=1, β∈HKq(A) and γ∈HKq(A), we
have the identities
Higher Koszul homology is the homology of the Koszul homology, and similarly for cohomology.
Precisely, let A=Tk(V)/(R) be a quadratic k-algebra over Q. Formula (6)
shows that the map eAK⌣eA:W2=R→A is zero. Therefore, eAK⌣− is a cochain differential on Homke(W∙,M), and eAK⌣− is a cochain differential on HK∙(A,M). Similarly, eAK⌢− is a chain differential
on M⊗keW∙, and eAK⌢− is a chain differential on HK∙(A,M). For a p-cocycle f:Wp→M and x1…xp+1 in Wp+1, we have
[TABLE]
For a p-cycle z=m⊗kex1…xp in M⊗keWp, we have
[TABLE]
{Df}
Let A=Tk(V)/(R) be a quadratic k-algebra over a finite quiver Q and let M be an A-bimodule. The differentials eAK⌣− and eAK⌢− are denoted by ∂⌣ and ∂⌢.
The homologies of the complexes (HK∙(A,M),∂⌣) and (HK∙(A,M),∂⌢) are called the higher Koszul cohomology and homology of A with coefficients in M and are denoted by HKhi∙(A,M) and HK∙hi(A,M). We set HKhi∙(A)=HKhi∙(A,A) and HK∙hi(A)=HK∙hi(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 HK∙(A), and ∂⌣(1)=eA implies that [eA]=0. If eA=0, the unit of HK∙(A) does not survive in higher Koszul cohomology.
As in the one vertex case, the actions of the Koszul cup and cap products of HK∙(A) on HK∙(A,M) and HK∙(A,M) induce actions on higher cohomology and homology. Thus HKhi∙(A) is a graded algebra, and HKhi∙(A,M), HK∙hi(A,M) are graded HKhi∙(A)-bimodules, constituting the higher Koszul calculus of A. If eA=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, eAK⌣− and eAK⌢− 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=Tk(V)/(R) be a quadratic k-algebra over Q and let M be an A-bimodule. Then HKhi0(A,M) is the space of elements u in Z(M) such that there exists v∈M satisfying
u.a=v.a−a.v for any a in Q1.
2.6 Grading the restricted Koszul calculus by the weight
A Koszul p-cochain f:Wp→Am is said to be homogeneous of weight m. Since Q1 is finite, the spaces Wp are finite dimensional, thus the space of Koszul cochains Homke(W∙,A) is
N×N-graded by the biweight(p,m), where p is called the homological weight and m is called the coefficient weight. If f:Wp→Am
and g:Wq→An are homogeneous of biweights (p,m) and (q,n) respectively, then fK⌣g:Wp+q→Am+n is homogeneous of biweight (p+q,m+n). Moreover bK is homogeneous of biweight (1,1) and the algebra HK∙(A)
is N×N-graded by the biweight. The homogeneous component of biweight (p,m) of HK∙(A) is denoted by HKp(A)m. Since
[TABLE]
the algebra HKhi∙(A) is N×N-graded by the biweight, and its (p,m)-component is denoted by HKhip(A)m. From Proposition 2.5, we deduce the following.
**Proposition **\theDf
Let A=Tk(V)/(R) be a quadratic k-algebra over Q. Assume that A is finite dimensional. Let max be the highest m such that Am=0. Then HKhi0(A)max is isomorphic to the space spanned by the cycles of Q of length max.
Similarly, a Koszul q-chain z in An⊗keWq is said to be homogeneous of weight n. The space of Koszul
chains A⊗keW∙ is N×N-graded by the biweight(q,n), where q is called the homological weight and n is called the
coefficient weight. Moreover bK is homogeneous of biweight (−1,1) and the space HK∙(A) is N×N-graded by the biweight.
The homogeneous component of biweight (q,n) of HK∙(A) is denoted by HKq(A)n. Since
[TABLE]
the space HK∙hi(A) is N×N-graded by the biweight, and its (q,n)-component is denoted by HKqhi(A)n.
If f:Wp→Am and z∈An⊗keWq are homogeneous of biweights (p,m) and (q,n) respectively,
then fK⌢z and zK⌢f are homogeneous of biweight (q−p,m+n) where
[TABLE]
and z=a⊗kex1…xq. The Homke(W∙,A)-bimodule A⊗keW∙, the HK∙(A)-bimodule HK∙(A) and the
HKhi∙(A)-bimodule HK∙hi(A) are thus N×N-graded by the biweight. The proof of the following is left to the reader.
**Proposition **\theDf
Let A=Tk(V)/(R) be a quadratic k-algebra over Q. We have
[TABLE]
Moreover HK0(A)1≅HK1(A)0 is isomorphic to the space spanned by the loops of Q, and ∂⌢:HK1(A)0→HK0(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=Tk(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 Q and Q′ be finite quivers, and F be a field. We introduce
the commutative rings k=FQ0 and k′=FQ′0, the
k-bimodule V=FQ1 and the k′-bimodule V′=FQ1′. As
explained in Subsection 2.1, we make the identifications of graded algebras Tk(V)≅FQ and Tk′(V′)≅FQ′. We are interested in the graded
F-algebra isomorphisms u:Tk(V)→Tk′(V′) given by a ring isomorphim
u0:k→k′ and by a k-bimodule isomorphism u1:V→V′, where V′ is a
k-bimodule via u0. By [25, Lemma 4], this implies that u0 maps Q0 to Q′0, and the bijection Q0→Q′0 induced by u0 transforms the adjacency matrix of Q into the adjacency matrix of Q′.
Let us fix a sub-k-bimodule R of V⊗kV and a sub-k′-bimodule R′ of V′⊗k′V′. We define the graded k-algebra A=Tk(V)/(R) and the graded k′-algebra A′=Tk′(V′)/(R′). Following the terminology of the one vertex case, a graded F-algebra isomorphism u:A→A′ is called a Manin isomorphism if u is defined by a ring isomorphism u0:k→k′ (so V′ is a k-bimodule via u0), and by a k-bimodule isomorphism u1:V→V′, such that the k-bimodule isomorphism u1⊗k2:V⊗k2→V′⊗k′2 satisfies u1⊗k2(R)=R′. In particular, u is an isomorphism of the augmented k-algebra A to the augmented k′-algebra A′, the augmentations being the projections A→A0≅k and A′→A0′≅k′.
As in [6], for any A-bimodule M, the Manin isomorphism u naturally defines an isomorphism of complexes from (M⊗keW∙,bK) to (M⊗k′eW∙′,bK), where M is an A′-bimodule via u, inducing natural isomorphisms HK∙(A,M)≅HK∙(A′,M). Similarly, u induces natural isomorphisms HK∙(A′,M)≅HK∙(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 u1(eA)=eA′ 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=Tk(V)/(R) over a finite quiver Q, naturally augmented over k by the
projection B→B0≅k. This implies that the ring C is isomorphic to k≅FQ0. Then we can define the (general) Koszul calculus of A as being the (general) Koszul calculus of B.
Indeed, if B′=Tk′(V′)/(R′) over a finite quiver Q′ is another quadratic quiver algebra presentation of A, the ungraded augmented k-algebra B is isomorphic to the ungraded augmented k′-algebra B′. By Gaddis’s theorem, there exists a Manin isomorphism from B to B′, thus the (general) Koszul calculi of B and B′ 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=Tk(V)/(R) be a quadratic k-algebra over Q. Recall that, for p=0 and p=1, we have linear isomorphisms HKp(A,M)≅HHp(A,M) and HHp(A,M)≅HKp(A,M) (Subsection 2.2). It is no longer true if p⩾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 HK2(A,M)→HH2(A,M) and an injection HH2(A,M)→HK2(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⊗kEp⊗kA, where Ep is a weight-graded k-bimodule. Then Pl(A)=P(A)⊗Ak (respectively Pr(A)=k⊗AP(A)) is a minimal projective resolution of the graded left (respectively right) A-module k (see for instance [5]) and the differential δ of P(A) is the graded sum of the differentials δ⊗Aidk and idk⊗Aδ naturally extended to P(A).
Define the left (respectively right) Koszul complex Kl(A)=K(A)⊗Ak (respectively Kr(A)=k⊗AK(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 ExtA(k,k) from the resolutions Pl(A) and Pr(A), we can show that Kl(A) (respectively Kr(A)) is isomorphic as a left (respectively right) A-module to the diagonal part of the graded resolution Pl(A) (respectively Pr(A)). We know (see for instance [40, Section 3]) that the differential d of K(A) is the graded sum of the differentials d⊗Aidk and idk⊗Ad naturally extended to K(A). Thus the inclusions A⊗kWp⊗kA↪A⊗kEp⊗kA constitute an inclusion map ι:K(A)↪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, E0=k, E1=V, E2=R, and the differential δ of P(A) coincides with the differential d of K(A) in degrees 1 and 2.
For any A-bimodule M, ι induces ι~=M⊗Aeι and ι∗=HomAe(ι,M) decomposed in ι~p:M⊗keWp→M⊗keEp and ιp∗:Homke(Ep,M)→Homke(Wp,M). The linear maps
[TABLE]
are isomorphisms for p=0 and p=1, and for any p if A is Koszul. Since ιp is an identity map for p=0,1,2, ι~p and ιp∗ are also identity maps for the same p’s. Therefore, setting δ~=M⊗Aeδ and δ∗=HomAe(δ,M), we have the commutative diagrams
[TABLE]
[TABLE]
where ι~3 is injective and ι3∗ is surjective (the ring ke≅F∣Q0∣2 is semisimple), so that we obtain the following.
**Proposition **\theDf
Let A=Tk(V)/(R) be a quadratic k-algebra over Q. For any M,
\edefitit(i)
H(ι~)2:HK2(A,M)→HH2(A,M)* is surjective with kernel
isomorphic to Im(δ~3)/Im(b3K),*
2. \edefitit(ii)
H(ι∗)2:HH2(A,M)→HK2(A,M)* is injective with image isomorphic to
Ker(δ3∗)/Im(bK2).*
We can be more specific when M=A, by using the weight grading (Subsection 2.6). Unlike the Koszul differentials bK and bK, the Hochschild differentials bH and bH are not homogeneous for the coefficient weight, but only for the total weight. The grading of HHp(A) and HHp(A) for the total weight t is denoted by HHp(A)t and HHp(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⊗ke(a1…ap) is equal to t=∣a∣+∣a1∣+…+∣ap∣, and the total weight of a homogeneous p-cochain f mapping a1…ap to an element of Am is equal to t=m−∣a1∣−…−∣ap∣. Then H(ι~)2 is homogeneous from the coefficient weight r to the total weight r+2, while H(ι∗)2 is homogeneous from the total weight r−2 to the coefficient weight r.
**Corollary **\theDf
Let A=Tk(V)/(R) be a quadratic k-algebra over Q.
\edefitit(i)
H(ι~)2* is an isomorphism from HK2(A)r to HH2(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 Am=0.
Then H(ι∗)2 is an isomorphism from HH2(A)r−2 to HK2(A)r if r=max and
r=max−1.
Proof 2.3**.**
Denote by Ep,m the homogeneous component of weight m of Ep. Since E3,2=0 and E3,3=W3, both maps δ~3 and b3K vanish on the component of total weight 2 of A⊗keE3, while on that of total weight 3, they coincide with the inclusion map of W3 into V⊗keR. Then we deduce (i) from (i) of the proposition.
Under the assumptions of (ii), if f:R→Amax, then bK3(f)=0. Moreover, any other component of δ3∗(f) mapping E3,m to Amax+m−2=0 vanishes as well. Thus δ3∗(f)=0, and we conclude by (ii) of the proposition. The same proof works if f:R→Amax−1 since δ3∗(f) is then reduced to a map W3→Amax coinciding with bK3(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 A~ defined just before Definition 2.3.
3.1 Compatibility
**Lemma **\theDf
Let A and B be unital associative F-algebras. Let M be an A-bimodule (hence the induced F-bimodule is symmetric). Assume that M is a right B-module such that the actions of F induced on M by A and by B are the same. Let Ae=A⊗FAop be the enveloping algebra. The following are equivalent.
\edefitit(i)
Viewing M as a left Ae-module, M is an Ae-B-bimodule.
2. \edefitit(ii)
Viewing M as a right Ae-module, the right actions of Ae 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=Ae, Ae is a natural Ae-Ae-bimodule for the multiplication of the F-algebra Ae. Recall that the left Ae-module Ae is isomorphic to the A-bimodule A⊗oA for the outer action (a⊗b).(α⊗β)=(aα)⊗(βb), while the right Ae-module Ae is isomorphic to the A-bimodule A⊗iA for the inner action (α⊗β).(a⊗b)=(αa)⊗(bβ).
3.2 DG bimodules over the DG algebra A~
Let A=Tk(V)/(R) be a quadratic k-algebra over Q. Fix a unital associative 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⊗keWq is a right B-module for the action of b∈B on z=m⊗kex1…xq∈M⊗keWq defined by
[TABLE]
It is well-defined since (λmμ).b=λ(m.b)μ for any λ and μ in k. From (4) and (7), we check that bK and eAK⌢− are B-linear. Thus HK∙(A,M) and HK∙hi(A,M) are graded right B-modules.
Just before Definition 2.3, we have associated to A the F-central DG algebra
[TABLE]
whose grading is given by the cohomological degree of cochains, whose differential is bK and whose multiplication is K⌣. We have also mentioned that Homke(W∙,M) and M⊗keW∙ are DG bimodules over A~ for the actions of K⌣ and K⌢ respectively, so that HK∙(A,M) and HK∙(A,M) are graded HK∙(A)-bimodules.
For any k-bimodule morphism f:Wp→A, we verify that
[TABLE]
so that the right action of B on M⊗keW∙ is compatible with the
A~-bimodule structure. Therefore the right action of B on HK∙(A,M) and on
HK∙hi(A,M) is compatible with the structure of HK∙(A)-bimodule and of
HKhi∙(A)-bimodule respectively. Let us sum up what we have obtained at the level of complexes.
**Proposition **\theDf
Let A=Tk(V)/(R) be a quadratic k-algebra over Q, let B be a unital associative
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 Mod-B the category of right B-modules. Then
\edefitit(i)
the complex (M⊗keW∙,bK) is a complex in Mod-B,
2. \edefitit(ii)
the complex (M⊗keW∙,bK) is a DG-bimodule over the
F-central DG-algebra A~,
3. \edefitit(iii)
the right action of B and the bimodule action of A~ on
M⊗keW∙ are compatible.
In this situation, following Yekutieli [45, Definition
3.8.1], we say that M⊗keW∙ is a DG
A~-bimodule in the abelian category Mod-B.
In order to reflect the fact that the right action of B and the bimodule
actions of HK∙(A) and HKhi∙(A) on HK∙(A,M) and
HK∙hi(A,M) respectively are compatible, we shall also say that HK∙(A,M) is a graded HK∙(A)-bimodule in
Mod-B, and that HK∙hi(A,M) is a graded HKhi∙(A)-bimodule in Mod-B. By Lemma 3.1, it is equivalent to saying that HK∙(A,M) is a graded HK∙(A)e-B-bimodule, similarly for HK∙hi(A,M).
Similarly, Homke(W∙,M) is a right B-module for the action of b on f:Wp→M defined by
[TABLE]
Then bK and eAK⌣− are B-linear, so that HK∙(A,M) and HKhi∙(A,M) are graded right B-modules. For g:Wq→A, we have
[TABLE]
We obtain an analogue of Proposition 3.2, that is, Homke(W∙,M) is a DG A~-bimodule in Mod-B.
**Proposition **\theDf
We keep the notation and assumptions of the previous proposition. Then
\edefitit(i)
the complex Homke(W∙,M) is a complex in Mod-B,
2. \edefitit(ii)
the complex Homke(W∙,M) is a DG A~-bimodule,
3. \edefitit(iii)
the right action of B and the bimodule action of A~ are compatible on
Homke(W∙,M).
Therefore HK∙(A,M) is a graded HK∙(A)-bimodule in Mod-B, and HKhi∙(A,M) is a graded HKhi∙(A)-bimodule in Mod-B.
3.3 Application to the Koszul complex K(A)
Let us specialise to B=M=Ae as in Example 3.1. Then M=A⊗oA is a left Ae-module for the outer structure, and a right Ae-module for the inner structure. Our aim is to identify the A-bimodule complex K(A) with the complex ((A⊗oA)⊗keW∙,bK) endowed with the right action of Ae. The statement is the following.
**Proposition **\theDf
Let A=Tk(V)/(R) be a quadratic k-algebra over Q.
\edefitit(i)
For any q⩾0, the bilinear map
φq:(A⊗oA)×Wq→A⊗kWq⊗kA defined
by
[TABLE]
induces an isomorphism
φ~q:(A⊗oA)⊗keWq→A⊗kWq⊗kA.
2. \edefitit(ii)
The direct sum φ~ of the maps φ~q is an isomorphism from the
complex ((A⊗oA)⊗keW∙,bK) to the Koszul complex
(K(A),d).
3. \edefitit(iii)
The isomorphism φ~ is right Ae-linear.
Proof 3.1**.**
The A-bimodule A⊗oA is a k-bimodule for the actions λ(α⊗β)μ=λα⊗βμ, with α and β in A, λ and μ in k, thus it is a right ke-module for (α⊗β)(λ⊗μ)=μα⊗βλ. Then it is easy to check that
[TABLE]
proving the existence of φ~q. We define similarly an inverse linear map, therefore
φ~q is an isomorphism, which gives (i).
Let us show that φ~ is a morphism of complexes. From
[TABLE]
we get
[TABLE]
whose right-hand side is equal to d(β⊗kx1…xp⊗kα), as expected.
Let us prove (iii). Here the A-bimodule A⊗kWq⊗kA is seen as a right Ae-module.
For z=(α⊗β)⊗kex1…xq and a, b in A, we have
[TABLE]
*therefore φ~q is Ae-linear.
*
So φ~ is an isomorphism from the A-bimodule complex ((A⊗oA)⊗keW∙,bK) whose A-bimodule structure is the inner one, to the A-bimodule complex K(A). Denote by A-Bimod the category of A-bimodules. According to Proposition 3.2, (A⊗oA)⊗keW∙ is a DG A~-bimodule in A-Bimod. We transport this structure via φ~ and we obtain.
**Proposition **\theDf
Let A=Tk(V)/(R) be a quadratic k-algebra over Q. Then the Koszul complex K(A) is a DG A~-bimodule in A-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 HK∙(A)-bimodule in A-Bimod, so that H(φ~):HK∙(A,A⊗oA)→H(K(A)) is an isomorphism of graded HK∙(A)-bimodules in A-Bimod.
Let us give explicitly the underlying A~-bimodule structure of the DG A~-bimodule K(A). Consider z=(α⊗β)⊗kex1…xq in (A⊗oA)⊗keWq and f in Homke(Wp,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
The differential eAK⌢− induces a differential, still denoted by ∂⌢, on H(K(A)). The homology of (H(K(A)),∂⌢) is denoted by Hhi(K(A)) and is called the higher homology of K(A). Then Hhi(K(A)) is a graded HKhi∙(A)-bimodule in A-Bimod and H(H(φ~)):HK∙hi(A,A⊗oA)→Hhi(K(A)) is an isomorphism of graded HKhi∙(A)-bimodules in A-Bimod.
4 Poincaré Van den Bergh duality of preprojective algebras
4.1 Preprojective algebras
Throughout this section, Δ 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 Δ 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 A~D~E~ [4, Definition 4.5.1].
Let Q be a quiver whose underlying graph is Δ. Define a quiver Q∗ whose vertex set is Q0 and whose arrow set is Q1∗={a∗;a∈Q1} where s(a∗)=t(a) and t(a∗)=s(a). Let Q be the double quiver of Q, that is, the quiver whose vertex set is Q0=Q0 and whose arrow set is the disjoint union Q1=Q1∪Q1∗. We shall view (−)∗ as an involution of Q1.
Let F be a field. As before, we denote the ring FQ0 by k and the k-bimodule FQ1 by V and we identify the graded k-algebras Tk(V)≅FQ
(see Subsection 2.1).
The preprojective algebra associated with the graph Δ over the field F is the quadratic k-algebra A(Δ) over Q defined by A(Δ)=FQ/(R), where the sub-k-bimodule R of FQ2 is generated by
[TABLE]
where ε(a)=1 if a∈Q1, ε(a)=−1 if a∈Q1∗.
If Q′ is another quiver whose underlying graph is Δ, and R′ is the sub-k-bimodule of FQ′2 generated by the relations σi′=∑a∈Q1′t(a)=iε(a)aa∗ for all i∈Q0′=Q0, then the preprojective algebras FQ/(R) and FQ′/(R′) 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(Δ) and the (general, higher) Koszul calculus of A(Δ) depend only on the graph Δ and not on Q, justifying the notation A(Δ). If Δ is a tree, A(Δ) is isomorphic to the preprojective algebra defined without signs (that is, ε(a)=1 for all a∈Q1, as in [18, 19, 20]).
If Δ=A1, then A(Δ)=k. If Δ=A2, then R=FQ2
and A(Δ)=FQ0⊕FQ1. 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 Δ is distinct from A1 and A2. The following are equivalent.
\edefitit(i)
Δ* is Dynkin of type ADE.*
2. \edefitit(ii)
A(Δ)* is not Koszul.*
3. \edefitit(iii)
A(Δ)* is finite dimensional.*
Proof 4.1**.**
The equivalence (i)⇔(ii) is treated in [33] if Δ is a tree, in [30] otherwise. The equivalence (i)⇔(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 eA(Δ) is not a coboundary if Δ has a loop or if charF=2 and Δ=A1. In the remainder of this section, we often abbreviate A(Δ) 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 Δ=A1, then K(A) has length 0. If Δ=A2, then K(A) has
infinite length. However, when Δ is not Dynkin ADE, the algebra A=A(Δ) has global
dimension 2 (this is a consequence of [10, Proposition 4.2], inspired by manuscript notes of
Crawley-Boevey), and since k≅F∣Q0∣ 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 Δ other than
A1 and A2, and we now give a unified proof of this.
**Theorem **\theDf
Let A=A(Δ) be a preprojective algebra over F with Δ=A1 and Δ=A2. Then the Koszul complex K(A) of A has length 2. Consequently, HKp(A,M)≅HKp(A,M)=0 for all A-bimodules M and all p⩾3.
Proof 4.2**.**
From the defining equality (2) of Wp, we have Wp=(Wp−1⊗kV)∩(V⊗kWp−1) for all p⩾3. Moreover R=0, therefore it is enough to prove that W3=0, that is, (R⊗kV)∩(V⊗kR)=0. For that, we only assume that Δ=A1. Our goal is to prove that W3=0 implies Δ=A2.
Let u be a non-zero element in W3, viewed as an element in FQ3. There exist vertices e,f in Q0 such that euf=0, therefore we may assume that u is in eW3f. Then u can be written uniquely as
[TABLE]
We now use the fact that Q1 is the disjoint union of Q1 and Q1∗ and the definition of ε to write
[TABLE]
From these expressions, we obtain the following identities in the path algebra FQ:
[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 FQ, where there are no relations between paths, the identities (1) to (4) above are equivalent to
[TABLE]
We have assumed that u=0, so that either there exists α∈eQ1f such that λα=0 or there exists α∈fQ1e such that λα∗=0, using the first expression of u. We separate the two cases.
Assume that there exists α∈eQ1f such that λα=0. Then it follows from identity (8) that Q1e is empty. From (5), for all a∈eQ1, there exist β∈eQ1f and b∈Q1f such that λαaa∗α=−μββb∗b. Hence β=a=b=α and therefore eQ1={α}=eQ1f and μβ=−λα=0. From (9), it follows that fQ1 is empty. Finally, (5) becomes
[TABLE]
so that ∑b∈Q1fb=ααb∗b=0 and hence Q1f={α}.
We have proved that Q1e=∅=fQ1 so that in particular e=f, and that Q1f={α}=eQ1=eQ1f. Finally, Q=e←αf and Δ=A2.
In the case where there exists α∈fQ1e such that λα∗=0, a similar proof using (7), (10) and (6) shows that Q=f←αe and Δ=A2.
As an immediate consequence of Proposition 2.4 and Theorem 4.2, we obtain that in the Koszul calculus of A(Δ), 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(Δ) be a preprojective algebra over F with Δ=A1 and Δ=A2. We consider an A-bimodule M. For any α∈HK∙(A,M), β∈HK∙(A) and γ∈HK∙(A), we have the identities
[TABLE]
The same conclusion holds if Δ=A1 (obvious) and if Δ=A2 (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 ω0∈A⊗keR defined for any graph Δ by
[TABLE]
From σi=∑a∈Q1,t(a)=iε(a)aa∗, we get
[TABLE]
Then it is easy to check that ω0 is a Koszul 2-cycle. Being homogeneous of weight 0, ω0 is not a 2-boundary whenever Δ=A1.
In the following statement, we need the DG algebra A~ and the DG A~-bimodules Homke(W∙,M) and M⊗keW∙, introduced just before Definition 2.3.
**Theorem **\theDf
Let A=A(Δ) be a preprojective algebra over F with Δ=A1 and Δ=A2. Consider the Koszul 2-cycle ω0=∑i∈Q0ei⊗σi∈A⊗keR. For each Koszul p-cochain f with coefficients in an A-bimodule M, we define the Koszul (2−p)-chain θM(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 θM:Homke(W∙,M)→M⊗keW2−∙ is an isomorphism of DG A~-bimodules.
It follows that H(θM):HK∙(A,M)→HK2−∙(A,M) is an isomorphism of graded HK∙(A)-bimodules and that H(H(θM)):HKhi∙(A,M)→HK2−∙hi(A,M) is an isomorphism of graded HKhi∙(A)-bimodules.
Proof 4.3**.**
First we show that fK⌢ω0=ω0K⌢f for all f∈Homke(Wp,M). Using the definition of ω0 and the equalities (13), (7) and (8), we obtain for p=0,1,2,
[TABLE]
Next, for f∈Homke(Wp,M) and g∈Homke(Wq,M), we have
[TABLE]
providing equalities (15). Therefore θM:Homke(W∙,M)→M⊗keW2−∙ is a morphism of graded A~-bimodules, where A~ is just considered as a graded algebra. It remains to examine what happens for the Koszul differentials.
Combining θN([eA,g]K⌣)=[eA,θN(g)]K⌢ with bK=−[eA,−]K⌣ and bK=−[eA,−]K⌢,
we deduce that θM is a morphism of complexes, thus a morphism of DG A~-bimodules.
We prove that θM is an isomorphism by giving an inverse map η:M⊗keW2−∙→Homke(W∙,M). We define ηp:M⊗keW2−p→Homke(Wp,M) for p=0,1,2, by
[TABLE]
where δ is the Kronecker symbol. It is routine to verify that these linear maps are well-defined and form an inverse map for θM.
Finally the isomorphism H(θM) of graded HK∙(A)-bimodules satisfies
[TABLE]
for all α∈HK∙(A,M). Therefore H(θM) is a morphism of complexes for higher (co)homologies. Taking higher (co)homologies, we get a HKhi∙(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 ω0, where ω0∈HK2(A)0. In the next subsection, we extract from this duality a generalisation of the 2-Calabi-Yau property.
Unless Δ has no loop and charF=2, the class eA∈HK1(A)1 is non-zero, hence
[TABLE]
is non-zero in HK1(A)1. Consequently, the fundamental class ω0 of the Poincaré Van den Bergh duality is not a cycle for the higher Koszul homology, so that the isomorphism H(H(θM)) cannot be naturally expressed as a cap action.
The class H(θA)(eA) is the class of the Koszul 1-cycle ω0K⌢eA where
[TABLE]
It is interesting to view the last element as the image by the canonical linear map can:V⊗kV→V⊗keV of the element
[TABLE]
In the identification V⊗kV≅FQ2, V⊗keV 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 σi, we can make the identification ω0K⌢eA=w. The element w was defined in [14, Proposition 8.1.1] as a representative of a bi-symplectic 2-form ω. 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 Δ=A2, so that A is defined by the quiver
[TABLE]
subject to the relations σ0=−a∗a, σ1=aa∗. Then the statement of Theorem 4.3 is valid in a weaker form, namely the isomorphisms involved are only morphisms. Moreover, θM is bijective only in degree q, 0≤q≤2, with the same inverse ηq. More generally, for any p≥1, the Koszul (2p)-cycle
[TABLE]
provides a morphism ωp−1K[]⌢− which is bijective only in degree q, 0≤q≤2p. From that, we deduce an isomorphism ωp−1K[]⌢− from HKq(A,M) to HK2p−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(Δ) be a preprojective algebra over F with Δ=A1 and Δ=A2. 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 Mod-B the category of right B-modules. Recall that A~ denotes the DG algebra (Homke(W∙,A),bK,K⌣).
According to Subsection 3.2, M⊗keW∙ and Homke(W∙,M) are DG A~-bimodules in Mod-B. Moreover, HK∙(A,M) and HK∙(A,M) are graded HK∙(A)-bimodules in Mod-B. Finally, HK∙hi(A,M) and HKhi∙(A,M) are graded HKhi∙(A)-bimodules in Mod-B.
**Lemma **\theDf
The map θM:Homke(W∙,M)→M⊗keW2−∙ is an isomorphism of DG A~-bimodules in Mod-B. Moreover, H(θM):HK∙(A,M)→HK2−∙(A,M) is an isomorphism of graded HK∙(A)-bimodules in Mod-B, and H(H(θM)):HKhi∙(A,M)→HK2−∙hi(A,M) is an isomorphism of graded HKhi∙(A)-bimodules in Mod-B.
Proof 4.4**.**
It is enough to prove that θM:f↦ω0K⌢f is B-linear. For a k-bimodule morphism f:Wp→M, z=a⊗kex1…xq∈A⊗keWq and b∈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=ω0, we obtain that θM is B-linear.
We specialise this lemma to M=B=Ae and, using the isomorphism φ~ in Subsection 3.3, we identify Ae⊗keW∙ with K(A) to get the next
proposition.
**Proposition **\theDf
Let A=A(Δ) be a preprojective algebra over F with Δ=A1 and Δ=A2. The map
[TABLE]
is an isomorphism of DG A~-bimodules in A-Bimod. Moreover,
[TABLE]
is an isomorphism of graded HK∙(A)-bimodules in A-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(Δ) be a preprojective algebra over F with Δ=A1 and Δ=A2. Then the HK∙(A)e-Ae-bimodules HK∙(A,Ae) and H2−∙(K(A)) are isomorphic. In particular, we have the following.
\edefitit(i)
The A-bimodule HK2(A,Ae) is isomorphic to the A-bimodule A.
2. \edefitit(ii)
HK1(A,Ae)=0.
3. \edefitit(iii)
The A-bimodule HK0(A,Ae) is isomorphic to the A-bimodule H2(K(A)).
Since H1(K(A))≅HK1(A,Ae)=0, the higher Koszul differentials vanish. Therefore Hphi(K(A))≅Hp(K(A)), HKhip(A,Ae)≅HKp(A,Ae) and H(H(θAe))≅H(θAe).
From the generator 1⊗k1 of the A-bimodule H0(K(A)), we draw from (i) a generator of the free A-bimodule HK2(A,Ae) defined as the class of f:R→A⊗oA with f(σi)=ei⊗ei for any i.
In (iii), the A-bimodules are never [math] when Δ is Dynkin of types 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 HHp(A,Ae) [28, §3.2]. In Ginzburg’s definition, HHp(A,Ae)=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=Tk(V)/(R)
over a finite quiver Q as defined in Subsection 2.1, and in the Koszul calculus of A as presented in the remainder of Section 2.
Note that Q1=Q1 if we want to specialise to preprojective algebras.
For the definition of the bounded derived category Db(C) of an abelian category C, we refer to [44, Chapter 10]. Recall that A-Bimod denotes the category of A-bimodules.
{Df}
Let A=Tk(V)/(R) be a quadratic k-algebra over Q. Let n⩾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
\edefnit(i)
the Koszul bimodule complex K(A) of A has length n, and
2. \edefnit(ii)
RHomAe(K(A),Ae)≅K(A)[−n] in Db(A-Bimod).
Property (ii) is equivalent to saying that there is an A-bimodule quasi-isomorphism from Homke(W∙,Ae) to K(A[−n]. According to Theorem 4.2 and Proposition 4.4, a preprojective algebra A(Δ) over F with Δ=A1 and Δ=A2 is Kc-Calabi-Yau of dimension 2. In fact, the isomorphism θAe induces an isomorphism RHomAe(K(A),Ae)→K(A)[−2] in Db(A-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 F-algebras.
{Df}
An associative 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) RHomAe(A,Ae)≅A[−n] in Db(A-Bimod).
Definition 5.1 is a true generalisation of Definition 5.1 for quadratic quiver algebras. If Δ is Dynkin of type ADE, A(Δ) is not Calabi-Yau in Ginzburg’s definition since A(Δ) is not homologically smooth in this case (the minimal projective resolution of A(Δ) has infinite length). However, the two definitions coincide if A is Koszul.
**Proposition **\theDf
Let A=Tk(V)/(R) be a quadratic k-algebra over 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)≅A in Db(A-Bimod). Thus RHomAe(A,Ae)≅A[−n] in Db(A-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)≅A in Db(A-Bimod), which allows us to conclude that A is n-Kc-Calabi-Yau.
If the graph Δ is not Dynkin ADE, we know that A(Δ) is Koszul (Proposition 4.1), thus we recover the fact that A(Δ) 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∈{0,1} 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∈{0,1}.
**Proposition **\theDf
Let A=Tk(V)/(R) be a quadratic k-algebra over Q. Then A is Calabi-Yau of dimension 0 if and only if Q1=∅.
We leave the proof as an exercise. If A is Calabi-Yau of dimension 1, then R=0, that is, A=Tk(V)≅FQ with Q1=∅. It is indeed 1-Calabi-Yau if Q has only one vertex and one loop, but we have not yet found other examples when Q is connected.
If A is n-Calabi-Yau, the Van den Bergh duality theorem states that the vector spaces HHp(A,M) and HHn−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 Q. Then for any A-bimodule M, the vector spaces HKp(A,M) and HKn−p(A,M) are isomorphic.
Proof 5.2**.**
Denote by VectF the category of F-vector spaces. For any A-bimodule M, the
left derived functor M⊗LAe− and the right derived functor RHomAe(−,M) are defined from Db(A-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↦HomAe(N,M) where M and N are seen as right Ae-modules. Specialising to M=Ae in F, we define a functor G:A-Bimod→A-Bimod. Let H:A\text{-{Bimod}}\rightarrow\text{{Vect}{}{\mathbb{F}}} be the functor H:N′↦M⊗AeN′ where N′ is viewed as a left Ae-module. Then we define a linear map
[TABLE]
by ϕM(m⊗Aeg)(x)=m.g(x) for m∈M, g∈HomAe(N,Ae) and x∈N. This map is functorial in N, defining a natural transformation ϕM:H∘G⇒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, ϕM 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
HKp(A,M)≅HKn−p(A,M) as vector spaces.
5.2 Koszul complex Calabi-Yau algebras versus Calabi-Yau algebras
Recall that if Δ=A1 and Δ=A2, then A(Δ) is 2-Kc-Calabi-Yau. But observe that if A(Δ) is moreover 2-Calabi-Yau, then A(Δ) is Koszul. We are led to the following conjecture.
{Cn}
Let A=Tk(V)/(R) be a quadratic k-algebra over 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=Tk(V)/(R) be a quadratic k-algebra over Q. Conjecture 5.2 holds if n⩽3.
Proof 5.3**.**
Assume that A is n-Calabi-Yau and n-Kc-Calabi-Yau. We can assume that n⩾2. Since HKp(A,Ae)≅HHp(A,Ae)=0 when p=0 and p=1, we have Hn(K(A))≅Hn−1(K(A))=0, hence A is Koszul if n=2. When n=3, we also have H1(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 Q. The image c∈HKn(A) of the unit 1 of the algebra A by the isomorphism HK0(A)≅HKn(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 A~=HomAe(K(A),A)) and we work with DG A~-bimodules in A-Bimod, as defined in Section 3 and recalled below. The point is that K(A) is such a DG A~-bimodule in A-Bimod (Proposition 3.3). Denote by C(A~,A-Bimod) and \mathcal{C}(\tilde{A},\text{{Vect}{}_{\mathbb{F}}}) the category of DG A~-bimodules in A-Bimod and VectF respectively [45]. Remark that a DG A~-bimodule in VectF is just a DG A~-bimodule.
{Df}
A DG A~-bimodule C in A-Bimod is a chain complex in A-Bimod (as usual, C can be viewed as a cochain complex) endowed with a DG A~-bimodule structure such that the bimodule actions of A and A~ are compatible.
For any A-bimodule M, HomAe(C,M) is a (cochain) DG A~-bimodule in VectF (in A-Bimod when M=Ae) for the following actions
[TABLE]
where f:A⊗kWp⊗kA→A, u:Cq→M and x∈Cp+q. Note that x.f and f.x are in Cq by the graded actions of A~ on C. If C=K(A), we recover the cup actions, that is, f.u=fK⌣u and u.f=uK⌣f. In particular, HomAe(K(A),Ae) is a DG A~-bimodule in A-Bimod.
Similarly, for any cochain DG A~-bimodule C′ in A-Bimod, M⊗AeC′ is a cochain DG A~-bimodule in VectF for the following actions
[TABLE]
where f∈A~, m∈M and u∈C′.
The bounded derived categories Db(A~,A-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 Cb(A~,A-Bimod) when M=Ae.
{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 HomAe(−,Ae) of Cb(A~,A-Bimod) exists and if RHomAe(K(A),Ae)≅K(A)[−n] in Db(A~,A-Bimod).
The preprojective algebras of connected graphs distinct from A1 and A2 are strong 2-Kc-Calabi-Yau algebras if they satisfy the first property in this definition. In fact, using Proposition 4.4, θAe provides then an isomorphism RHomAe(K(A),Ae)→K(A)[−2] in Db(A~,A-Bimod).
**Theorem **\theDf
Let A be a Kc-Calabi-Yau algebra of dimension n over Q, with fundamental class c. We assume that A is strong Kc-Calabi-Yau and that the derived functors of the functors HomAe(−,A) and A⊗Ae− from Cb(A~,A-Bimod) to \mathcal{C}^{b}(\tilde{A},\text{{Vect}{}_{\mathbb{F}}}) exist. Then
[TABLE]
is an isomorphism of HK∙(A)-bimodules, inducing an isomorphism of HKhi∙(A)-bimodules from HKhi∙(A) to HKn−∙hi(A). For all α∈HKp(A), we have cK⌢α=(−1)npαK⌢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 A~-bimodule in
A-Bimod. We prove now that ϕM is a morphism of DG 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 ϕM is a morphism of A~-bimodules. Let us prove that ϕM is left A~-linear, the right linearity being similar. For f:A⊗kWp⊗kA→A, u:Cq→Ae and x∈Cp+q, we have
[TABLE]
while f.(ϕM(m⊗Aeu))(x)=(−1)pϕM(m⊗Aeu)(x.f)=(−1)pm.(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 Cb(A~,A-Bimod) to \mathcal{C}^{b}(\tilde{A},\text{{Vect}{}_{\mathbb{F}}}), and H is now an endofunctor of Cb(A~,A-Bimod). Under these notations, ϕM defines a natural transformation ϕM:H∘G⇒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 ϕA [45]. Then for any bounded chain complex DG A~-bimodule C in A-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 HK∙(A)≅HKn−∙(A) of graded bimodules over the graded algebra H(A~)=HK∙(A). Denote this isomorphism by ψ.
The fact that ψ is a morphism of graded HK∙(A)-bimodules translates as
[TABLE]
for any α∈HKp(A) and β∈HK∙(A). In accordance with Definition 5.3, define c∈HKn(A) by c=ψ(1) where 1∈HK0(A) is the unit of A. Applying identities (3) to the trivial equalities α=1K⌣α=αK⌣1, we obtain
[TABLE]
Finally ψ is a morphism of complexes for higher (co)homologies since we have
[TABLE]
Then H(ψ):HKhi∙(A)→HKn−∙hi(A) is an isomorphism of HKhi∙(A)-bimodules.
Except in some particular cases, eA∈HK1(A) is non-zero, so that
[TABLE]
is non-zero in HKn−1(A). Therefore c∈HKn(A) is not a cycle for higher Koszul homology and the isomorphism H(ψ) cannot be naturally expressed as a cap action. As suggested by the preprojective algebras (Subsection 4.3), the class ψ(eA) should be of interest for further investigations.
It is also interesting to remark that the identities (3) involving the isomorphism ψ imply that the graded algebra HK∙(A) is commutative if and only if the graded HK∙(A)-bimodule HK∙(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.
(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
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∈HKp(A) and x∈HKq(A), we have
[TABLE]
3. (N3)
Let X and Y be N-graded spaces and let f:X→Y be a homogeneous
map of degree 1. Let y1,…,yp be elements of pairwise different degrees. Then if
∑i=1pyi∈Imf, at least one of the yi is in Imf. We shall use this in the
following context. The differentials bK1 and bK2 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 κ:A→A constructed as follows.
Let A be a preprojective algebra over a graph Δ; let Q be its quiver. Consider the map
Q1→Q1 that sends a to a∗. It induces an anti-automorphism κ of A such that κ(ei)=ei for all i∈Q0
(since κ sends the relation σi=∑a∈Q1t(a)=iε(a)aa∗ to itself).
5. (N5)
We shall be using the Nakayama automorphism ν 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 ν on the set of vertices of Δ. It is known that ν=id if Δ is Dn with n even or if Δ is E7 or E8, and that otherwise ν is induced by the unique graph automorphism of order 2.
If α is an arrow in Q1, let β be the unique arrow from ν(s(α)) to ν(t(α)). The Nakayama automorphism of [10] is described as follows:
[TABLE]
Given a basis of a selfinjective quiver algebra A=FQ/I consisting of paths and containing a
basis {πi;i∈Q0} 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 ν is the
Nakayama automorphism corresponding to this bilinear form (characterised on the arrows α in Q1
by yα=πs(α)⇔ν(α)y=πt(α) for all the basis elements y).
6. (N6)
When we define a cochain f∈Homke(X,A) with X∈{k,V,A}, it will be implicit
that if f(x) is not defined for some x∈X then f(x)=0.
7. (N7)
For any cochain f∈Homke(Wp,A), we shall set fˇ=θA(f)∈A⊗keW2−p.
8. (N8)
Finally, given a Dynkin graph Δ and a ring L, we shall denote by ΛL the
preprojective algebra of Δ over L, so that A=ΛF.
6.1 Koszul calculus for preprojective algebras of type A
The preprojective algebra A of type An is defined by the quiver
[TABLE]
subject to the relations
[TABLE]
The Nakayama automorphism ν of A described in 5 is given by ν(ei)=en−1−i, ν(ai)=an−2−i∗ and ν(ai∗)=an−2−i.
Erdmann and Snashall have given in [18] a basis B of A. We shall only need the sets eiBei, eiBei+1 and ei+1Bei, which can be rewritten as follows:
set mA=⌊2n−1⌋; then
[TABLE]
For each i, Aei 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 HK0(A)=HH0(A)=Z(A) and HK1(A)=HH1(A) are known from [18].
Therefore we only need to compute HK2(A). Recall our assumption that n⩾3; then by
Theorem 4.2 all the elements in Homke(R,A) are
cocycles. Moreover, using Theorem 4.3
and [24, Theorem 3.2.7], we have dimHK2(A)=dimHK0(A)=dimHH0(A)=n. Since every element in ImbK2 has coefficient weight at least 1, the n cocycles hi defined by hi(σj)=δijei for all j are linearly independent modulo ImbK2. It follows that they form a basis of HK2(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 HK0(A) is given by the set {zℓ;0⩽ℓ⩽mA} with z0=1 and
zℓ=∑i=1n−2(ai∗ai)ℓ=∑i=ℓn−1−ℓ(ai∗ai)ℓ=z1ℓ for
1⩽ℓ⩽mA.
A basis of HK1(A) is given by the set
{ζℓ;0⩽ℓ⩽n−2−mA}, where ζℓ∈Homke(V,A) is defined by
ζℓ(ai)=ai(ai∗ai)ℓ for all i (or for ℓ⩽i⩽n−2−ℓ).
A basis of HK2(A) is given by the set
{hi;0⩽i⩽n−1} where hi∈Homke(R,A) defined by hi(σj)=δijei for all j.
As a consequence of Theorem 4.3, we obtain
bases of the Koszul homology spaces.
**Corollary **\theDf
A basis of HK0(A) is given by the set
{hˇi;0⩽i⩽n−1} where hˇi=ei⊗ei.
A basis of HK1(A) is given by the
set {ζˇℓ;0⩽ℓ⩽n−mA−2} where ζˇℓ=∑i=0n−2ai(ai∗ai)ℓ⊗ai∗=∑i=ℓn−2−ℓai(ai∗ai)ℓ⊗ai∗.
A basis of HK2(A) is given by the set
{zˇℓ;0⩽ℓ⩽mA} where zˇℓ=∑i=0n−1(ai∗ai)ℓ⊗σi.
Note that zˇ0=ω0 is the fundamental class.
6.1.2 Cup and cap products
We know from Corollary 4.2 that the cup product on HK∙(A) is graded-commutative. The following
result gives all the non zero cup products of elements in HK∙(A).
**Proposition **\theDf
Let A be a preprojective algebra of type An.
Up to graded commutativity, the non zero cup products in HK∙(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 HH0(A) and HH1(A), therefore they are known from [18].
The basis elements of HK2(A) have coefficient weight [math], and bK2 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.
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 eA is equal to the cohomology class of 2ζ0.
Proof 6.2**.**
Let ζ0∗∈Homke(V,A) be the cocycle defined by ζ0∗(ai∗)=ai∗
for all i∈Q0.
Since eA=ζ0+ζ0∗, we must prove that ζ0∗−ζ0 is a coboundary.
Consider v=∑i=0n−2∑j=in−2ei∈⨁i∈Q0eiAei≅Homke(k,A). Then bK1(v)=ζ0∗−ζ0, as required.
As a consequence, the complex defining the higher Koszul cohomology is
[TABLE]
with ∂⌣1(zℓ)=2ζℓ for 0⩽ℓ⩽n−mA−2 and ∂⌣2=0. We then have the following higher Koszul cohomology.
**Proposition **\theDf
Let A be a preprojective algebra of type An.
If char(F)=2, then HKhi∙(A)=HK∙(A).
If char(F)=2 and n is even, then
[TABLE]
Finally, if char(F)=2 and n is odd, then
[TABLE]
Higher Koszul homology can then be deduced using duality (Theorem 4.3).
**Corollary **\theDf
If char(F)=2, then HK∙hi(A)=HK∙(A).
If char(F)=2 and n is even, then
[TABLE]
Finally, if char(F)=2 and n is odd, then
[TABLE]
6.2 Koszul calculus for preprojective algebras of type D
The preprojective algebra A of type Dn is defined by the quiver
[TABLE]
subject to the relations
[TABLE]
The Nakayama automorphism ν of A described in 5 is given by ν(ei)=ei, ν(ai)=−ai∗ and ν(ai∗)=ai if n is even of if n is odd and i⩾2, and it exchangess e0 and e1, a0 and −a1, and a0∗ and a1∗ if n is odd.
Eu has given in [22] a basis B of A. Set mD=⌊2n−2⌋ and u=n−mD−2.
We shall only need bases of the ejAei when i and j are equal or adjacent vertices, which can be rewritten as follows:
[TABLE]
For each i, Aei 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 F by fact 1 and was computed in
[22] in characteristic [math]. A basis of HH1(ΛC) was determined in [22]. Moreover, dimHK1(A)=dimHH1(A), which is equal to dimHH1(ΛC)=n
if char(F)=2 and to dimHH1(ΛC)+m if char(F)=2 by [24].
It also follows from Theorem 4.3 and [24] that
dimHK2(A)=dimHK0(A)=dimHH0(A), which is equal to n−mD−2 if char(F)=2 and to
n−2 if char(F)=2.
In order to give bases of the HKp(A) for p=0,1,2, we define the following cochains:
•
the elements z0=1 and
zℓ=(a0∗a1a1∗a0)ℓ+(a1∗a0a0∗a1)ℓ+∑i=2n−2(ai∗ai)2ℓ=z1ℓ for
ℓ>0 in A. Note that if n is even, then
z1u=z1mD=−π0+π1, but if n is odd then z1u=0;
•
the elements ζℓ∈Homke(V,A) with 0⩽ℓ⩽u−1 defined by ζℓ(ai)=aizℓ for all i;
•
the elements ρℓ∈Homke(V,A) with 0⩽ℓ⩽mD−1 where
ρℓ(ai)=(a2∗a2)2ℓ+1ai for i=0,1 and ρℓ(ai∗)=ai∗(a2∗a2)2ℓ+1 for i=0,1;
•
the elements hj∈Homke(R,A) for
0⩽j⩽n−1, where hj(σi)=δijej;
•
the elements γℓ∈Homke(R,A) for 1⩽ℓ⩽mD where γℓ(σ0)=(a0∗a1a1∗a0)ℓ.
**Proposition **\theDf
Let A be a preprojective algebra of type Dn.
\edefitit(i)
The elements in {πi;0⩽i⩽n−1 and ν(ei)=ei}∪{zℓ;0⩽ℓ⩽u−1}
form a basis of HK0(A).
2. \edefitit(ii)
If char(F)=2, the ζℓ, for 0⩽ℓ⩽u−1,
form a basis of HK1(A).
If char(F)=2, the ζℓ for 0⩽ℓ⩽u−1 and the
ρℓ for 0⩽ℓ⩽mD−1 form a basis of HK1(A).
3. \edefitit(iii)
If char(F)=2, the hj for
0⩽j⩽n−1 form a basis of HK2(A).
If char(F)=2, the hj for
0⩽j⩽n−1 and the γℓ for 1⩽ℓ⩽mD form a basis of HK2(A).
Proof 6.3**.**
The results for HK0(A) and, when char(F)=2, for HK1(A) follow from the comments before the proposition.
Assume that char(F)=2. In order to prove the result for HK1(A), we must prove that the elements we have considered in Homke(V,A)
are cocycles that are linearly independent modulo coboundaries.
It is in fact enough to prove that the ρℓ are cocycles that are not coboundaries by fact
3.
First note that, at the level of cochains, ρℓ=ρ0K[]⌣zℓ. Therefore, to prove that
ρℓ is a cocycle, it is enough to prove that ρ0 is a cocycle, and this is easy to check.
Since ρℓK[]⌣zmD−1−ℓ=ρmD−1, in order to prove that ρℓ is not a coboundary for all ℓ, it is enough to prove that ρmD−1 is not a coboundary. The map ρmD−1
has coefficient weight 4mD−1. If ρmD−1 is a coboundary, then it is the image of a
morphism in Homke(k,A)≅⨁i∈Q0eiAei whose coefficients are linear
combinations of cycles in A of weight 4mD−2, which are known. It is then straightforward to show that the image of any such morphism under bK1 is not
equal to ρmD−1.
For (iii), we first observe that every cochain in Homke(R,A) is a
cocycle. Moreover, the hj are n cocycles that are clearly linearly independent modulo coboundaries (all coboundaries have coefficient weight at least equal to 1). Therefore if char(F)=2, the result follows.
If char(F)=2, it is enough to prove that the γℓ are not coboundaries by fact
3. At the level of cochains, γℓK[]⌣zmD−ℓ=γmD for
1⩽ℓ⩽mD, therefore it is enough to prove that γmD is not a coboundary. The map γmD has
coefficient weight 4mD, therefore if γmD is a coboundary, then it is the image of a
morphism in Homke(V,A) whose coefficients are linear
combinations of elements of weight 4mD−2 in A between two adjacent vertices in Q0, which are known. Here again, checking that that the image of any such morphism under bK2 is not
equal to γmD 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⩽i⩽n−2, consider the cochains ui,vi and wi in Homke(R,A) defined by
ui(σj)=δijeiz1,
vi(σj)=δijπi and
wi(σj)=δijai∗aiz1 for all j∈Q0.
If char(F)=2, the ui, vi and wi are all coboundaries.
If char(F)=2, then the ui for i⩾2 are coboundaries, and u0=u1=γ1. Moreover, if n is odd, all
the vi are coboundaries and if n is even, then vi=γmD for all i.
Finally, the wi are all coboundaries.
Proof 6.4**.**
Every element in Homke(R,A) is a cocycle. Moreover, the differential bK2 is homogeneous of degree 1 with respect to the coefficient weight, and the coefficient weight of all the basis elements in HK2(A) is a multiple of 4, and is [math] if char(F)=2.
It follows that if char(F)=2, all the ui,vi and wi must be coboundaries, and if char(F)=2, the wi are coboundaries and so are the vi if n is odd.
Assume that char(F)=2. We must now study the ui, as well as the vi when n is even.
Note that un−2=0 and u0=γ1. For 0⩽i⩽n−3, define pi∈Homke(V,A) by
pi(a0)=a1a1∗a0, pi(a1)=a0a0∗a1 and pi(ai)=aiai∗ai if i⩾2.
Then, for 2⩽i⩽n−3, we have ui=bK2(∑j=in−3pj). Moreover, bK2(∑j=0n−3pj)=u0+u1.
It follows that the cohomology classes of u0 and u1 are both equal to that of γ1.
We now turn to the vi. Note that since n is even and char(F)=2, the map v0 is the map γmD, which is not a coboundary.
Define qi∈Homke(V,A) by qi(ai)=(ai+1∗ai+1)n−i−2ai⋯a2a1a1∗a2∗⋯ai−1∗ for 1⩽i⩽n−2 and
q0(a0)=(a2∗a2)n−3. Then, for 2⩽i⩽n−2, we have
vi−v1=bK2(∑j=1i−1qj), and
v1−v0=bK2(q0−q1). Therefore vi=v0=γmD 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 HK∙(A) are:
For ℓ⩾1, we have zℓK[]⌣hi=zℓ−1K[]⌣z1K[]⌣hi=zℓ−1K[]⌣ui and the result follows from Lemma 6.2.2.
Next, πiK[]⌣hj=δijvj and again the result is a consequence of Lemma 6.2.2.
Now assume that char(F)=2, so that the ρℓ occur in the basis of HK1(A).
At the level of cochains, we have
ρℓ1K[]⌣ρℓ2=w2K[]⌣zℓ1+ℓ2, which is a coboundary.
The map ρℓ1K[]⌣ζℓ2=u2K[]⌣zℓ1+ℓ2+1 is also a coboundary, as
required.
The remaining cup products are easy to compute. Note that the cup product in HK0(A)≅Z(A) is
the ordinary product, and that the elements πi 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 2ζ0 so that
∂⌣1(zℓ)=2ζℓ for 0⩽ℓ⩽u−1, ∂⌣1(πi)=0 and ∂⌣2=0. We then have the following higher Koszul cohomology.
**Proposition **\theDf
Let A be a preprojective algebra of type Dn.
\edefitit(i)
If char(F)=2, then HKhi∙(A)=HK∙(A).
2. \edefitit(ii)
If char(F)=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 E6 is defined by the quiver
To simplify notation, we shall denote by c0=a0a0∗, c2=a2a2∗ and c3=a3∗a3 the three
2-cycles at the vertex 3.
The socle is the part of weight 10 of A, and the set {πi;i∈Q0} where
π0=a0∗c32c0c3a0, π1=a4a3c0c3c0a2a1, π2=a2∗(c3c0)2a3∗,
π3=c3(c0c3)2, π4=κ(π2) and π5=κ(π1).
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: z0=1,
z6=a1∗a2∗a3∗a3a2a1+a2∗c32a2−c0c3c0+a3c22a3∗+a4a3a2a2∗a3∗a4∗ and
z8=a2∗c0c3c0a2+c0c32c0+a3c0c3c0a3∗;
•
in Homke(V,A): the maps ζℓ defined by ζℓ(ai)=aizℓ for
ℓ∈{0,6,8}, the map ρ3 defined by ρ3(a2)=c0a2, ρ3(a3)=a3c3 and
ρ3(a2∗)=a2∗c3, and the map ρ5 defined by ρ5(a0)=c2c3a0, ρ5(a1)=a2∗c0a2a1,
ρ5(a2)=c22a2, ρ5(a0∗)=−a0∗c32, ρ5(a1∗)=−a1∗a2∗c0a2 and
ρ5(a2∗)=a2∗c32;
•
in Homke(R,A): the maps hj defined for 0⩽j⩽5 by hj(σi)=δijej for
all i, the map γ4 defined by γ4(σ0)=a0∗c3a0 and the map γ6 defined by γ6(σ0)=a0∗c32a0.
We shall use the following lemma.
**Lemma **\theDf
Assume that char(F)=3. Let γ∈Homke(R,A) be an element of coefficient weight 6,
so that
[TABLE]
Then γ is a coboundary if, and only if, ∑i=05λi+∑i=24λi′+λ3′′=0.
Proof 6.6**.**
The proof is straightforward, once we know that a cochain of weight 5 takes its values in
A5=E⊕κ(E), where E is the space spanned by c32a0, c0c3a0, a2∗c0a2a1,c32a2, c0c3a2, a3c0c3, a3c3c0, a4a3c0a3∗.
**Proposition **\theDf
Let A be a preprojective algebra of type E6.
\edefitit(i)
The elements in {z0,z6,z8,π0,π3} form a basis of HK0(A).
2. \edefitit(ii)
If char(F)∈{2,3}, the elements in
{ζℓ;ℓ=0,6,8} form a basis of HK1(A).
If char(F)=2, the elements in {ζℓ;ℓ=0,6,8}∪{ρ3} form a basis of HK1(A).
If char(F)=3, the elements in
{ζℓ;ℓ=0,6,8}∪{ρ5} form a basis of HK1(A).
3. \edefitit(iii)
If char(F)∈{2,3}, the elements in {hj;j∈Q0} form a basis of HK2(A).
If char(F)=2, the elements in {hj;j∈Q0}∪{γ4} form a basis of HK2(A).
If char(F)=3, the elements in {hj;j∈Q0}∪{γ6} form a basis of HK2(A).
For HK1(A) and HK2(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 char(F) is not 2 or 3, a basis of HK1(A)=HH1(A) was given in [22]. It
consists of the classes of the ζℓ′ with ℓ∈{0,6,8} where
ζℓ′(ai)=aizℓ for 0⩽i⩽2 and ζℓ′(ai∗)=ai∗zℓ for 3⩽i⩽4. Since ζ0−ζ0′ is equal to bK1(e3+2e5), and
ζℓ−ζℓ′=(ζ0−ζ0′)K[]⌣zℓ is also a coboundary, ζℓ and
ζℓ′ represent the same cohomology class for ℓ∈{0,6,8}. Moreover, as in types A and D, the elements hj form a basis of HK2(A).
If char(F)∈{2,3}, we need only prove that the extra elements are not coboundaries by fact
3.
If char(F)=2, we have z6K[]⌣ρ3−ζ8=bk1(g) where g is defined by g(e2)=a2∗c3c0c3a2, and ζ8 is not a coboundary,
therefore ρ3 cannot be a coboundary. Moreover, assume that γ4=bK1(g′) is a coboundary. Then g′ would be of coefficient weight 3,
and we would necessarily take values in A3=E⊕κ(E) where E is spanned by c3a0,
c0a2,c3a2,
a3c0,a3c3. This leads to a contradiction.
If char(F)=3, assume that ρ5 is a coboundary
bk1(h), then h is of weight 4, and necessarily h(e0)=λ0a0∗c3a0,
h(e2)=λ2a2∗c3a2 and h(e3)=λ3c3c0+λ3′c32+λ3′′c0c3, and by considering bK1(h)(a0), bK1(h)(a0∗), bK1(h)(a2) and bK1(h)(a2∗) we get
a contradiction.
Finally, the fact that γ6 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 HK∙(A) are:
[TABLE]
Proof 6.8**.**
The first two cup products are clear.
For the cup products of z6 with the hi and for ζ0K[]⌣ρ5, we use Lemma 6.3.1.
The last cup product follows from the fact that we have ζ0K[]⌣ρ3−γ4=bK1(g′) where g′(a0∗)=a0∗c3 and g′(a3)=a3c3. The cup product
z6K[]⌣ρ3 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 ∂⌣1 sends zℓ to 2ζℓ for
ℓ∈{0,6,8} and the differential ∂⌣2 is zero.
We then have the following higher Koszul cohomology.
6.4 Koszul calculus for preprojective algebras of type E7
The preprojective algebra A of type E7 is defined by the quiver
[TABLE]
subject to the relations
[TABLE]
The Nakayama automorphism is given by ν(ai)=−ai and ν(ai∗)=ai∗ for i∈Q0.
To simplify notation, we shall denote by c0=a0a0∗, c2=a2a2∗ and c3=a3∗a3 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
π0=(a0∗c3a0)4, π1=−a1∗a2∗c0c3(c3c0)2a2a1, π2=−(a2∗c0a2)4,
π3=(c3c0)3c32, π4=−(a3c0a3∗)4, π5=−a4a3(c3c0)3a3∗a4∗ and π6=−a5a4(a3c0a3∗)3a4∗a5∗.
6.4.1 The Koszul cohomology and homology spaces in type E7
We define the following elements
•
in A: z0=1,
z8=a0∗c2c0c2a0−a2∗c2c0c2a2−c2c32c2+a3c0c2c0a3∗−a4a3c22a3∗a4∗+a5a4a3c0a3∗a4∗a5∗ and
z12=a0∗(c2c0)2c2a0∗+a2∗(c0c2)2c0a2−(c3c0c3)2+a3c3(c0c3)2a3∗;
•
in Homke(V,A): the maps ζℓ defined by ζℓ(ai)=aizℓ for
ℓ∈{0,8,12}, the map ρ3 defined by ρ3(a2)=c0a2, ρ3(a3)=a3c3, ρ3(a4)=a4a3a3∗ and
ρ3(a2∗)=a2∗c3, the map ρ7 defined by ρ7(a0)=c33a0+c3c0c3a0, ρ7(a3)=a3c3c0c3 and ρ7(a3∗)=c3c0c3a3∗, the map ρ15 defined by ρ15(a0)=(c2c0)3c2a0 and ρ15(a0∗)=a0∗c2(c0c2)3 and the map ρ5 defined by ρ5(a0)=−c2c3a0, ρ5(a2)=c3c0a2, ρ5(a3)=a3c32, ρ5(a0∗)=a0∗c32 and ρ5(a2∗)=−a2∗c2c0;
•
in Homke(R,A): the maps hj defined for 0⩽j⩽6 by hj(σi)=δijej for
all i, the map γ4 defined by γ4(σ0)=a0∗c3a0, the map γ8 defined by γ8(σ0)=a0∗c33a0, the map γ16 defined by γ16(σ0)=π0 and the map and γ6 defined by γ6(σ0)=a0∗c32a0.
**Lemma **\theDf
First assume that char(F)=2.
\edefitit(i)
Let u16∈Homke(R,A) be an element of weight 16 so that u16(σi)=λiπi for i∈Q0. Then u16 is a coboundary if, and only if, ∑i=06λi=0.
2. \edefitit(ii)
Let u8∈Homke(R,A) be an element of weight 8 so that u8(σ0)=λ0a0∗c33a0, u8(σ2)=λ2a2∗c3c0c3a2+λ2′a2∗c32c0a2, u8(σ3)=λ3c33c0+λ3′ccc33+λ3′′c0c32c0+λ3′′′c32c0c3, u8(σ4)=λ4a3c32c0a3∗+λ4′a3c3c0c3a3∗+λ4′′a3c0c32a3∗, u8(σ5)=λ5a4a3c3c0a3∗a4∗+λ5′a4a3c0c3a3∗a4∗ and u8(σ6)=a5a4a3a0a0∗a3∗a4∗a5∗. Then u8 is a coboundary if, and only if, λ0+λ2+λ2′+λ3+λ3′+λ3′′′+λ4+λ4′+λ4′′+λ5+λ5′+λ6=0.
Now assume that char(F)=3.
(iii)
Let u6∈Homke(R,A) be an element of weight 6 so that u6(σ0)=λ0a0∗c32a0, u6(σ1)=λ1a1∗a2∗c0a2∗a1∗, u6(σ2)=λ2a2∗c3c0a2+λ2′a2∗c32a2, u6(σ3)=λ3c3c0c3+λ3′c32c0+λ3′′c0c3c0+λ3′′′c33+, u6(σ4)=λ4a3c0c3a3∗+λ4′a3c3c0a3∗ and u6(σ5)=λ5a4a3c0a3∗a4∗. Then u6 is a coboundary if, and only if, ∑i=05λi−λ2′+λ3′−λ4′=0.
Proof 6.9**.**
For each ℓ∈{16,6}, if the map uℓ were a coboundary, it would be the image of
a map gℓ∈Homke(V,A) whose coefficients would be in the space generated by the paths
between adjacent vertices with weight ℓ−1. The proof is then straightforward once we know
bases of these spaces. Note that once we have a basis of ⨁α∈Q1et(α)Awes(α), applying κ gives a basis of
es(α)Awet(α) for a given weight w.
In weight 15, a basis of ⨁α∈Q1et(α)A15es(α) is given by a0=(a0∗c3a0)3a0∗c3, a1=−a1∗a2∗c0c3(c3c0)2a2, a2=−a2∗c0(c2c0)3, a3=(c3c0)3c3a3∗, a4=−a3(c3c0)3a3∗a4∗, a5=a4(a3c0a3∗)3a4∗a5∗.
In weight 7, a basis of ⨁α∈Q1et(α)A7es(α) is given by c33a0, c3c0c3a0, a2∗c32a2a1, c32c0a2, c0c32a2, c3c0c3a2, a3c3c0c3, a3c0c32, a3c32c0. a4a3c3c0a3∗, a4a3c0c3a3∗, a5a4a3c0a3∗a4∗.
In weight 5, a basis of ⨁α∈Q1et(α)A5es(α) is given by c32a0, c0c3a0, a2∗c0a2a1,
c3c0a2, c0c3a2, a3c3c0, a3c0c3, a3c32, a4a3c0a3∗.
**Proposition **\theDf
Let A be a preprojective algebra of type E7.
\edefitit(i)
The elements in {z0,z8,z12}∪{πi;i∈Q0} form a basis of HK0(A).
2. \edefitit(ii)
If char(F)∈{2,3}, the elements in
{ζℓ;ℓ=0,8,12} form a basis of HK1(A).
If char(F)=2, the elements in {ζℓ;ℓ=0,8,12}∪{ρ3,ρ7,ρ15} form a basis of HK1(A).
If char(F)=3, the elements in
{ζℓ;ℓ=0,8,12}∪{ρ5} form a basis of HK1(A).
3. \edefitit(iii)
If char(F)∈{2,3}, the elements in {hj;j∈Q0} form a basis of HK2(A).
If char(F)=2, the elements in {hj;j∈Q0}∪{γ4,γ8,γ16} form a basis of HK2(A).
If char(F)=3, the elements in {hj;j∈Q0}∪{γ6} form a basis of HK2(A).
For HK1(A) and HK2(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 char(F) is not 2 or 3, a basis of HK1(A)=HH1(A) was given in [22]. It
consists of the classes of the ζℓ′ with ℓ∈{0,8,12} where
ζℓ′(ai)=aizℓ for 0⩽i⩽2 and ζℓ′(ai∗)=ai∗zℓ for 3⩽i⩽5. Since ζ0−ζ0′ is equal to bK1(e3+2e5+3e6), and
ζℓ−ζℓ′=(ζ0−ζ0′)K[]⌣zℓ is also a coboundary, ζℓ and
ζℓ′ represent the same cohomology class for ℓ∈{0,8,12}. Moreover, as in types
A, D and E6, the elements hj form a basis of HK2(A).
If char(F)∈{2,3}, we need only prove that the extra elements are not coboundaries by fact
3.
If char(F)=2, it follows from Lemma 6.4.1 that ζ0K[]⌣ρ15−γ16, ζ8K[]⌣ρ7−γ16, ζ12K[]⌣ρ3−γ16,
z8K[]⌣γ8−γ16 and z12K[]⌣γ4−γ16 are coboundaries. Therefore it
is enough to check that γ16 is not a coboundary, and this also follows from Lemma
6.4.1.
If char(F)=3, again using Lemma 6.4.1, ζ0K[]⌣ρ5−γ6
is a coboundary and γ6 is not a coboundary, therefore ρ5 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 HK∙(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∈Homke(V,A3) is defined by h(a0)=c3a0 and h(a2)=c2a2.
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 ∂⌣1 sends zℓ to 2ζℓ for
ℓ∈{0,8,12} and the differential ∂⌣2 is zero except when char(F)=3 where ∂⌣2(ρ5)=γ6.
We then have the following higher Koszul cohomology.
6.5 Koszul calculus for preprojective algebras of type E8
The preprojective algebra A of type E8 is defined by the quiver
[TABLE]
subject to the relations
[TABLE]
The Nakayama automorphism is given by ν(ai)=−ai and ν(ai∗)=ai∗ for i∈Q0.
To simplify notation, we shall denote by c0=a0a0∗, c2=a2a2∗ and c3=a3∗a3 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 π0=(a0∗c2a0)7, π1=a1∗a2∗(c0c2)5c2c0a2a1, π2=−(a2∗c0a2)7, π3=(c2c0)7, π4=−a3(c2c0)6c2a3∗, π5=a4a3(c2c0)6a3∗a4∗, π6=a5a4a3(c0c2)5c0a3∗a4∗a5∗ and π7=−a6a5a4a3(c0c2)3(c2c0)2a3∗a4∗a5∗a6∗.
6.5.1 The Koszul cohomology and homology spaces in type E8
z20=a5a4a3(c0c2)3c0a3∗a4∗a5∗+a4a3(c0c2)2(c2c0)2a3∗a4∗+a3c0(c2c0)4a3∗+(c2c0)5−(c0c22)3c0+(c0c2)5+a2∗(c2c0c2)3a2+a0∗(c2c0)4c2a0
and
4. ✦
z24=z122;
2. ∙
in Homke(V,A):
✦
the maps ζℓ defined by ζℓ(ai)=aizℓ for
ℓ∈{0,12,20,24},
2. ✦
the map ρ3 defined by ρ3(a2)=c0a2, ρ3(a3)=a3c3, ρ3(a4)=a4a3a3∗, ρ3(a5)=a5a4a4∗ and
ρ3(a2∗)=a2∗c3,
3. ✦
the map ρ7 defined by ρ7(a0)=c0c32a0, ρ7(a3)=a3c0c32+a3c32c0+a3c0c3c0 and ρ7(a3∗)=c3c0c3a3∗,
4. ✦
the map ρ15 defined by ρ15(a3)=a3c2(c2c0)2, ρ15(a4)=a4a3(c0c3)2a3∗+a4a3(c0c32)2a3∗+a4a3(c3c0)3a3∗, ρ15(a5)=a5a4a3c0(c3c0)2a3∗a4∗, ρ15(a0∗)=a0∗c3(c3c0)3 and ρ15(a5∗)=a4a3(c0c3)2c0a3∗a4∗a5∗,
5. ✦
the map ρ27 defined by ρ27(a3)=a3c0(c3c0)6 and ρ27(a3∗)=(c3c0)6c3a3∗,
6. ✦
the map ρ5 defined by ρ5(a0)=−c0c3a0−c32a0, ρ5(a3)=a3c3c0+a3c32, ρ5(a4)=a4a3c3a3∗, ρ5(a0∗)=a0∗c32 and ρ5(a3∗)=−c3c0a3∗,
7. ✦
the map ρ17 defined by defined by
ρ17(a0)=−c32(c0c3)3a0−(c0c3)4a0,
ρ17(a3)=a3(c3c0)4+a3(c0c3)4+a3(c3c0)3c32,
ρ17(a0∗)=a0∗(c2c0)3c32 and ρ17(a3∗)=−(c3c0)4a3∗, and
8. ✦
the map ρ9 defined by ρ9(a0)=−2c32c0c3a0+2c3c0c32a0+(c0c3)2a0, ρ9(a2)=c22c0c2a2+(c2c0)2a2, ρ9(a3)=−a3(c0c3)2, ρ9(a0∗)=2a0∗c32c0c3−2a0∗c3c0c32+a0∗(c3c0)2, ρ9(a2∗)=a2∗c2c0c22−a2∗(c2c0)2 and ρ9(a3∗)=(c0c3)2a3∗;
3. ∙
in Homke(R,A): the maps hj defined for 0⩽j⩽7 by hj(σi)=δijej for
all i, and
[TABLE]
**Lemma **\theDf
First assume that char(F)=2.
\edefitit(i)
Let u28∈Homke(R,A) be an element of weight 28, so that
u28(σi)=λiπi for all i∈Q0. Then
u28 is a coboundary if, and only if, ∑i∈Q0λi=0.
Now assume that char(F)=3.
(ii)
Let u18∈Homke(R,A) be an element of weight 18, so that u18(σ0)=λ0a0∗c32(c0c3)3a0+λ0′a0∗c3c0(c3c0c3)2a0,
u18(σ1)=λ1a1∗a2∗c0(c2c0)3a2a1,
u18(σ2)=λ2a2∗(c2c0c2)2c0c2a2+λ2′a2∗(c0c2)4a2+λ2′′a2∗(c2c0)4a2,
u18(σ3)=λ3c0(c3c0)4+λ3′(c3c0)4c3+λ3′′(c3c0)2c3(c3c0)2+λ3(3)(c0c3)2c3(c0c3)2+λ3(4)(c3c0)3c32c0+λ3(5)c0c32(c0c3)3,
u18(σ4)=λ4a3(c3c0)4a3∗+λ4′a3(c0c3)4a3∗+λ4′′a3c32(c3c0)3a3∗+λ4(3)a3(c0c3)3c32a3∗, u18(σ5)=λ5a4a3(c3c0)3c3a3∗a4∗+λ5′a4a3c3(c3c0)3a3∗a4∗+λ5′′a4a3(c0c3)3c3a3∗a4∗,
u18(σ6)=λ6a5a4a3(c3c0)3a3∗a4∗a5∗+λ6′a5a4a3(c0c3)3a3∗a4∗a5∗,
u18(σ7)=λ7a6a5a4a3(c0c3)2c0a3∗a4∗a5∗a6∗. Then u18 is a
coboundary if, and only if, λ0+λ0′+λ1+λ2′+λ2′′+λ3′+λ3′′+λ3(3)+λ3(4)+λ3(5)+λ4+λ4′−λ4′′−λ4(3)−λ5−λ5′−λ5′′−λ6−λ6′−λ7=0.
2. (iii)
The map ρ17∈Homke(V,A) is not a coboundary.
Now assume that char(F)=5.
(iv)
Let u10∈Homke(R,A) be an element of weight 10, so that
u10(σ0)=λ0a0∗c32c0c3a0+λ0′a0∗c3c0c32a0,
u10(σ1)=λ1a1∗a2∗c0c2c0a2a1,
u10(σ2)=λ2a2∗(c0c2)2a2+λ2′a2∗(c2c0)2a2+λ2′′a2∗c0c22c0a2,
u10(σ3)=λ3c0(c3c0)2+λ3′c3(c0c3)2+λ3′′c3(c3c0)2+λ3(3)(c0c3)2c3+λ3(4)c3c0c32c0+λ3(5)c0c32c0c3,
u10(σ4)=λ4a3(c0c3)2a3∗+λ4′a3(c3c0)2a3∗+λ4′′a3c3c0c32a3∗+λ4(3)a3c32c0c3a3∗,
u10(σ5)=λ5a4a3c0c3c0a3∗a4∗+λ5′a4a3c32c0a3∗a4∗+λ5′′a4a3c0c32a3∗a4∗, u10(σ6)=λ6a5a4a3c3c0a3∗a4∗a5∗+λ6′a5a4a3c0c3a3∗a4∗a5∗,
u10(σ7)=λ7a6a5a4a3c0a3∗a4∗a5∗a6∗. Then u10 is a coboundary
if, and only if, λ0+λ0′+λ1+λ2+λ2′+λ2′′+λ3′+λ3′′+λ3(3)+λ3(4)+λ3(5)+λ4−λ4′+2λ4′′+2λ4(3)+λ5+2λ5′+2λ5′′+2λ6+2λ6′+2λ7=0.
Proof 6.12**.**
For each ℓ∈{28,18,10}, if the map uℓ were a coboundary, it would be the
image of a map gℓ∈Homke(V,A) whose coefficients would be in the space generated by
the paths between adjacent vertices with weight ℓ−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 a0=a0∗c2(c0c2)6, a1=a1∗a2∗(c0c2)5c32a2, a2=−a2∗c0(c2c0)6, a3=(c2c0)6c2a3∗, a4=−a3(c2c0)6a3∗a4∗, a5=−a4a3c0(c2c0)5a3∗a4∗a5∗, a6=a5a4a3(c0c2)3(c2c0)2a3∗a4∗a5∗a6∗,
and the κ(ai) for all i∈Q0.
In weight 17, the space ⨁α∈Q1et(α)A18es(α) has basis
(c0c3)4a0, c32(c0c3)3a0, c3c0(c3c0c3)2a0, a2∗c2c0c2(c2c0)2a2a1,
a2∗(c0c2)3c0a2a1, (c3c0)4a2, (c0c3)4a2, (c0c3)3c3c0a2, c3c0(c3c0c3)2a2, a3(c3c0)4, a3(c0c3)4, a3(c0c3)3c3c0,
a3c0c3(c3c0)3, a3(c3c0)3c32, a4a3(c0c3)3c0a3∗,
a4a3(c0c3)3c3a3∗, a4a3c3(c3c0)3a3∗, a5a4a3(c3c0)3a3∗a4∗,
a5a4a3(c0c3)3a3∗a4∗, a6a5a4a3(c0c3)2c0a3∗a4∗a5∗ and
⨁α∈Q1es(α)A18et(α)=κ(⨁α∈Q1et(α)A18es(α)).
In weight 9, the space ⨁α∈Q1et(α)A9es(α) has basis
c32c0c3a0, c3c0c32a0, (c0c3)2a0, a2∗c0c2c0a2a1, c22c0c2a2,
(c2c0)2a2, (c0c2)2a2, c0c22c0a2, a3c3c0c32, a3(c0c3)2,
a3(c3c0)2, a3c0c32c0, a4a3c32c0a3∗, a4a3c0c32a3∗,
a4a3c0c3c0a3∗, a5a4a3c3c0a3∗a4∗, a5a4a3c0c3a3∗a4∗, a6a5a4a3c0a3∗a4∗a5∗ and
⨁α∈Q1es(α)A9et(α)=κ(⨁α∈Q1et(α)A9es(α)).
Finally, if ρ17 were a coboundary, it would be equal to bk1(g) for some
g∈Homke(k,A16). Such a map g would necessarily satisfy
g(e0)∈span{a0∗(c3c0)3c3a0}, g(e3)∈span{(c3c0)4,(c0c3)4,c0c3(c3c0)3,(c0c3)3c3c0,c32(c0c3)3,c3c0(c3c0c3)2}, and
g(e4)∈span{a3c0(c3c0)3a3∗,a3c3(c0c3)3a3∗,a3c3(c3c0)3a3∗,a3(c0c3)3c3a3∗}.
Then,
by considering bk1(g)(a0), bk1(g)(a0∗) and bk1(g)(a3) we get a contradiction.
**Proposition **\theDf
Let A be a preprojective algebra of type E8.
\edefitit(i)
The elements in {z0,z12,z20,z24}∪{πi;i∈Q0} form a basis of HK0(A).
2. \edefitit(ii)
If char(F)∈{2,3,5}, the elements in
{ζℓ;ℓ=0,12,20,24} form a basis of HK1(A).
If char(F)=2, the elements in {ζℓ;ℓ=0,12,20,24}∪{ρ3,ρ7,ρ15,ρ27} form a basis of HK1(A).
If char(F)=3, the elements in
{ζℓ;ℓ=0,12,20,24}∪{ρ5,ρ17} form a basis of HK1(A).
If char(F)=5, the elements in
{ζℓ;ℓ=0,12,20,24}∪{ρ9} form a basis of HK1(A).
3. \edefitit(iii)
If char(F)∈{2,3,5}, the elements in {hj;j∈Q0} form a basis of HK2(A).
If char(F)=2, the elements in {hj;j∈Q0}∪{γ4,γ8,γ16,γ28} form a basis of HK2(A).
If char(F)=3, the elements in {hj;j∈Q0}∪{γ6,γ18} form a basis of HK2(A).
If char(F)=5, the elements in {hj;j∈Q0}∪{γ10} form a basis of HK2(A).
For HK1(A) and HK2(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 char(F) is not 2, 3 or 5, a basis of HK1(A)=HH1(A) was given in [22]. It
consists of the classes of the ζℓ′ with ℓ∈{0,12,20,24} where
ζℓ′(ai)=aizℓ for 0⩽i⩽2 and ζℓ′(ai∗)=ai∗zℓ for 3⩽i⩽6. Since ζ0−ζ0′ is equal to bK1(e3+2e5+3e6+4e7), and
ζℓ−ζℓ′=(ζ0−ζ0′)K[]⌣zℓ is also a coboundary, ζℓ and
ζℓ′ represent the same cohomology class for ℓ∈{0,12,20,24}. Moreover, as in types
A, D, E6 and E7, the elements hj form a basis of HK2(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: ζ0K[]⌣ρ27−γ28, ζ12K[]⌣ρ15−γ28,
ζ24K[]⌣ρ3−γ28, ζ20K[]⌣ρ7−γ28,
z24K[]⌣γ4−γ28, z12K[]⌣γ16−γ28,
z20K[]⌣γ8−γ28, z12K[]⌣ρ5−ρ17,
z12K[]⌣γ6−γ18, ζ0K[]⌣ρ9−γ10,
whereas γ28, γ18, γ10 and ρ17 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 HK∙(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):
z24K[]⌣ρ3=ρ27+bK1([e3↦c0(c3c0)6]), we have z12K[]⌣ρ3=ρ15+bK1(h)
where h is defined by h(a2)=(c0c2)3c0a2,
h(a3)=a3c0(c3c0)3+a3c3(c0c3)3, h(a4)=a4a3(c0c32)2a3∗ and
h(a2∗)=a2∗c0(c2c0)3+a2∗c2(c0c2)3, and finally z12K[]⌣ρ9+ζ20=bK(h′) where h′ is defined by h′(e0)=a0∗(c3c0)4c3a0, h′(e2)=2a2∗(c2c0)4c2a2, h′(e3)=−2(c0c3)5+2c32(c0c3)4+(c3c0)4c32, h′(e4)=−a3(c0c3)4c0a3∗+a3c3(c3c0)4a3∗−a3(c0c3)4c3a3∗ and h′(e5)=−a4a3(c0c3)4a3∗a4∗.
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 ∂⌣1 sends zℓ to 2ζℓ for
ℓ∈{0,8,12} and the differential ∂⌣2 is zero except when char(F)=5 where ∂⌣2(ρ9)=2γ10.
We then have the following higher Koszul cohomology.
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∙,∂∙) 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(ι∗)2 sends the Hochschild cohomology
class of an element in Ker(∂3∘−) to its Koszul cohomology class, and the surjection
H(ι~)2 induces an isomorphism between HH2(A) and
HK2(A)/Im(idA⊗∂3).
We first transport the maps ∂3∘− and idA⊗∂3 via the natural isomorphisms A⊗Ae(Aej⊗eiA)→eiAej that sends λ⊗(a⊗b) to bλa and HomAe(Aei⊗ejA,A)≅eiAej that sends f to f(ei⊗ej).
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 ei, i∈Q0 and a basis {πi;i∈Q0} of the socle of A, and such that each v∈B belongs to ejAei for some i,j in Q0.
Then if x∈Aei, (y,x) is the coefficient of πi in the expression of yx as a linear combination of elements in B.
The Nakayama automorphism ν of A satisfies (y,x)=(ν(x),y) for all x,y in A, and
induces a permutation of the indices, the Nakayama permutation ν, that is, a permutation of Q0 such that top(Aei)≅soc(Aeν(i)), characterised by ν(ei)=eν(i).
Let B be the dual basis of B with respect to the non degenerate form (−,−), so that (w,v)=δvw for all v,w in B. In particular, if v∈Bei, the coefficient of πi in vv is 1. Note that v∈ejBei if and only if v∈eν(i)Bej.
Then the maps ∂3∘− and
idA⊗Ae∂3 become respectively
[TABLE]
It then follows as in [24, Proposition 3.2.25] that Imδ3 is the span of
the δ3(ei) such that ν(i)=i, and that for such an i we have
δ3(ei)=∑j∈Q0ν(j)=jtr(ν∣ejAei)πj. Moreover, the matrix whose coefficients are the
tr(ν∣ejAei) 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 Kerδ3 identifies with
the kernel of the Cartan matrix of A. Moreover, for any element of positive weight a∈A, we
have δ3(a)=0. Therefore HH2(A) is obtained by taking all the elements of positive
weight in HK2(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 HH2(A) with HK2(A) and HH2(A) with HK2(A) in each case. In particular, the injection HH2(A)→HK2(A) is not surjective except in type E8 with char(F)=2.
6.6.1 Comparison of the second Koszul and Hochschild cohomology groups
In type A, the space HH2(A) was completely described by Erdmann and Snashall in [18], and they
proved that dimHH2(A)=n−mA−1 and gave a basis {h~i;0⩽i⩽n−mA−2} with
h~i=hi+hn−1−i. The morphism of complexes ι2∗ sends h~i to hi+hn−1−i, and this describes the injection HH2(A)→HK2(A).
In type D, if char(F)=2 and n is even, there is nothing to do since
HH2(A)=0. If char(F)=2 and n is odd, then dimHH2(A)=1, the basis given in
[22] for HH2(ΛC) also gives a basis of HH2(A), and it is the
cohomology class of the map ψ0 defined by ψ0(σ0)=e0 and ψ0(σ1)=−e1. The embedding HH2(A)→HK2(A) is therefore given by ψ0↦h0−h1.
Now assume that char(F)=2. Then dimHH2(A)=n+mD−2. As we explained above, a basis of
HH2(A) may be obtained from a basis of the set of elements of positive coefficient weight in HK2(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 HH2(A) is given by
[TABLE]
where ψ2p+1(σ2p+1)=e2p+1, ψ2p+2(σ2p+2)=e2p+2 and
ψ2p+2(σ2)=e2, φ2p(σ2p)=e2p,
φ2p+1(σ2p+3)=e2p+3 and φ2p+1(σ3)=e3, for p⩾1.
Therefore, the embedding HH2(A)→HK2(A) fixes the γℓ and sends
φ2p to h2p and φ2p+3 to h3+h2p+3 when n is even, and ψ0 to
h0+h1, ψ2p+1 to h2p+1 and ψ2p+2 to h2+h2p+2 when n is odd.
In type E6,
the Cartan matrix is equivalent, through row operations, to
[TABLE]
Let φ0, φ1, φ2 and φ3 be the maps in Homke(R,A)
defined by φ0(σ0)=e0,
φ1(σ1)=e1, φ1(σ5)=−e5, φ2(σ2)=e2,
φ2(σ4)=−e4 and φ3(σ3)=e3. Then HH2(A) is the subspace of HK2(A) spanned by
[TABLE]
In type E7, if char(F)=2, we have dimHH2(A)=dimHK2(A)>0, so that HH2(A) is
precisely the subspace of HK2(A) of elements of positive weight (which is zero unless char(F)∈{2,3}). If char(F)=2, then the
Cartan matrix of A is equivalent, through row operations, to the matrix (1000101) so that HH2(A) is the subspace of HK2(A) spanned by γ4, γ8, γ16,
h1, h2, h3, h5, h0+h4 and h0+h6.
In type E8, we only need to look at dimensions. Indeed, if
char(F)=2, then dimHH2(A)=dimHK2(A) so that HH2(A)≅HK2(A), and if
char(F)=2 then dimHH2(A)=dimHK2(A)>0 so that HH2(A) is
precisely the subspace of HK2(A) of elements of positive weight (which is zero unless char(F)∈{2,3,5}).
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 char(F) divides (mA+1), we have seen that dimHK2(A)=dimHH2(A) and therefore HK2(A)≅HH2(A). Now assume that n is odd and char(F)∤(mA+1). Then dimHK2(A)=dimHH2(A)+1. We must determine the image of the map δ3 given in Subsection 6.6.
The Nakayama permutation ν has precisely one fixed point, which is mA. The matrix of ν restricted to emAAemA is the identity matrix ImA+1.
Consequently, δ3(emA)=(mA+1)πmA spans the image of δ3.
Via the isomorphism ⨁i∈Q0eiAei≅A⊗keR, πmA corresponds to
zˇmA so that we have HH2(A)≅HK2(A)/span{zˇmA}.
In type D, if n is odd and char(F)=2, then we know that dimHK2(A)=dimHH2(A) and therefore HK2(A)≅HH2(A). In the other cases, we need to determine the image of the map δ3. We should note that the Nakayama automorphism in [22] differs from ν by composition with the inner automorphism given by the invertible element u=−e0+∑i=1n−1(−1)iei (after changing the labelling of vertices and arrows so that they are the same as ours).
If n is odd and char(F)=2, the fixed points of the Nakayama permutation are the integers
i with 2⩽i⩽n−1. The matrix (n−2)×(n−2) matrix Hν whose (i,j)-coefficient is the trace of ν
restricted to ejAei 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
tr(ν∣ejAei)=2 if i and j are even and is equal to [math] otherwise.
Therefore for all i fixed by ν, we have δ3(ei)=∑p=1mD+12π2p. Using the isomorphism
⨁i∈Q0eiAei≅A⊗keR, we obtain HH2(A)≅HK2(A)/span{∑p=1mD+1πˇ2p}.
If n is even, all the integers i with 0⩽i⩽n−1 are fixed points of ν. The matrix
Hν was given in [22, paragraph 11.2.2] and the same adaptations as in the case n odd gives the n×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 HH2(A) and HK2(A) are isomorphic.
Finally, in type E7, if char(F)=3, then HH2(A)≅HK2(A). Now assume that char(F)=3. The matrix
Hν was given in [22]; here again, the Nakayama automorphism in [22] differs from ν by composing with the inner automorphism associated with the element −e0+e1−e2+e3+e4+e5+e6 (after change of labelling) and therefore the non zero rows of the matrix Hν are those corresponding to vertices
0,4 and 6 and are all equal to (3000303).
It follows that HH2(A)≅HK2(A)/span{πˇ0+πˇ4+πˇ6}.
6.7 A minimal complete list of cohomological invariants
**Theorem **\theDf
Let A be the preprojective algebra of a Dynkin graph Δ over F. Assume that A has type either An with n⩾3, or Dn with n⩾4, or En with n=6,7,8. Let A′ be a preprojective algebra of type ADE, where the integer n′ concerning A′ is subjected to the same assumptions. Denote by (dp) the equality dimHKhip(A)=dimHKhip(A′). If (dp) holds for p=0,1 and 2, then n=n′, and A and A′ have the same type. The conclusion of this implication does not hold if (d2) 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)
Assume that A and A′ have types A or D. If charF=2, then n=n′
by (d2), and A and A′ have the same type by (d0). If charF=2 and A and if A′ both
have type A, then n=n′ by (d2). If charF=2, A is of type An and A′ is of type
Dn′, then the sum of (d0) and (d1) shows that n=2n′+mD′−2 or −3 which contradicts
(d2): n=n′+mD′. If charF=2 and if A and A′ both have type D, then n=n′ by (d1).
2. (2)
If A and A′ have type E, then n=n′ by (d0).
3. (3)
If A is of type An and A′ of type E6, then (d2) implies either
that n=6 if charF=2,3 or that n=7 if charF=2,3, but each case is excluded by (d0). Similarly when A′ is of type E7 and E8.
4. (4)
If A is of type Dn and A′ is of type E6, then (d2) implies one of
the three following cases: n=6 if charF=2,3, n=7 if charF=3 or n+mD=7 if
charF=2, but each case is excluded by (d0). Similarly when A′ is of type E7 and
E8 (we also use (d1) for E8).
Let us show that we cannot remove assumption (d2). It is clear if charF=2 because when
A is of type A3 and A′ is of type A5, both (d0) and (d1) hold. If charF=2, we
check that when A is of type A9 and A′ is of type E6 then (d0) and (d1) hold.
It is obvious that HKhi∙(A)≅F if A is of type A1 and it is easy to check that HKhi∙(A)=0 when A is of type A2, 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 charF=2, the product of the algebra HKhi∙(A) is identically zero. If charF=2, HKhi∙(A)=HK∙(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.
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