Deleting or adding arrows of a bound quiver algebra and Hochschild (co)homology

TL;DR

This paper investigates how the Hochschild (co)homology of bound quiver algebras varies with modifications to the quiver, such as adding or deleting arrows, using advanced homological tools.

Contribution

It provides a detailed description of the changes in Hochschild (co)homology under quiver modifications employing relative Hochschild (co)homology and the Jacobi-Zariski sequence.

Findings

Describes the impact of arrow modifications on Hochschild (co)homology.

Utilizes the Jacobi-Zariski long exact sequence for analysis.

Employs a one-step relative projective resolution of tensor algebras.

Abstract

We describe how the Hochschild (co)homology of a bound quiver algebra changes when adding or deleting arrows to the quiver. The main tools are relative Hochschild (co)homology, the Jacobi-Zariski long exact sequence obtained by A. Kaygun and a one step relative projective resolution of a tensor algebra.

Equations

$$\dots \stackrel{d}{\to }{P}_2\stackrel{d}{\to }{P}_1\stackrel{d}{\to }{P}_0\stackrel{d}{\to }M\to 0$$

$$0\to {\mathsf{Hom}}_A(P_0,N)\to {\mathsf{Hom}}_A(P_1,N)\to \cdots$$

$$\cdots \to N{\otimes }_A{P}_1\to N{\otimes }_A{P}_0\to 0$$

$${\mathsf{H}}^{\ast }(A|B,M)={\mathsf{Ext}}_{(A-A)|(B-B)}^{\ast }(A,M)$$

$${\mathsf{H}}_{\ast }(A|B,M)={\mathsf{Tor}}_{\ast }^{(A-A)|(B-B)}(M,A).$$

$$T=T_B(N)=B\oplus N\oplus N{\otimes }_{B}N\oplus \cdots \oplus N^{{\otimes }_{B}n}\oplus \cdots$$

$${\mathsf{H}}^{\ast }(T|B,X)=0,$$

$${\mathsf{H}}_{\ast }(T|B,X)=0.$$

$$0\to T{\otimes }_{B}N{\otimes }_{B}T\stackrel{g}{\to }T{\otimes }_{B}T\stackrel{f}{\to }T\to 0$$

$$N^{{\otimes }_{B}i}{\otimes }_B{N}^{{\otimes }_{B}j}\subset {\left(T{\otimes }_{B}T\right)}_n=\underset{k+l=n}{⨁}\left(N^{{\otimes }_{B}k}{\otimes }_B{N}^{{\otimes }_{B}l}\right)\subset T{\otimes }_{B}T.$$

$$(\stackrel{▽}{,}{x}_1,\dots ,x_n)\in B{\otimes }_B{N}^{{\otimes }_{B}n}\subset (T{\otimes }_{B}T{)}_n,$$

$$(x_1,\dots ,x_n\stackrel{▽}{,})\in N^{{\otimes }_{B}n}{\otimes }_{B}B\subset (T{\otimes }_{B}T{)}_n.$$

$$\begin{array}[]{lll}N^{\otimes_{B}i}\otimes_{B}N\otimes_{B}N^{\otimes_{B}j}&\subset(T\otimes_{B}N\otimes_{B}T)_{n}\\ &=\bigoplus_{k+1+l=n}\left(N^{\otimes_{B}k}\otimes_{B}N\otimes_{B}N^{\otimes_{B}l}\right)\\ &\subset T\otimes_{B}N\otimes_{B}T.\end{array}$$

$$\stackrel{▽}{x_1},\dots ,x_n\in B{\otimes }_{B}N{\otimes }_B{N}^{{\otimes }_B(n-1)}\subset (T{\otimes }_{B}N{\otimes }_{B}T{)}_n,$$

$$x_1,\dots ,\stackrel{▽}{x_n}\in N^{{\otimes }_B(n-1)}{\otimes }_{B}N{\otimes }_{B}B\subset (T{\otimes }_{B}N{\otimes }_{B}T{)}_n.$$

$$f(x_1,\dots ,x_i,\stackrel{▽}{,}{x}_{i+1},\dots ,x_n)=x_1,\dots ,x_n$$

$$\begin{array}[]{lll}g(x_{1},\dots,x_{i},\stackrel{{\scriptstyle\triangledown}}{{x_{i+1}}},x_{i+2},\dots,x_{n})=(x_{1},\dots,x_{i+1}\stackrel{{\scriptstyle\triangledown}}{{,}}\dots,x_{n})-(x_{1},{\dots}\stackrel{{\scriptstyle\triangledown}}{{,}}x_{i+1},\dots,x_{n}).\end{array}$$

$$0⟶(T{\otimes }_{B}N{\otimes }_{B}T{)}_n\stackrel{g}{⟶}(T{\otimes }_{B}T{)}_n\stackrel{f}{⟶}{T}_n⟶0$$

$$r_i:N^{{\otimes }_{B}i}{\otimes }_B{N}^{{\otimes }_{B}j}\to (T{\otimes }_{B}N{\otimes }_{B}T{)}_n$$

$$r_i(x_1,\dots ,x_i,\stackrel{▽}{,}{x}_{i+1},\dots ,x_n)=\sum _{k=1}^i(x_1,\dots ,\stackrel{▽}{x_k},\dots ,x_n).$$

$$\begin{array}[]{llll}rg(x_{1},\dots,\stackrel{{\scriptstyle\triangledown}}{{x_{i+1}}},\dots,x_{n})=\\ r_{i+1}(x_{1},\dots,x_{i+1}\stackrel{{\scriptstyle\triangledown}}{{,}}\dots,x_{n})-r_{i}(x_{1},{\dots}\stackrel{{\scriptstyle\triangledown}}{{,}}x_{i+1},\dots,x_{n})=\\ \sum_{k=1}^{i+1}(x_{1},{\dots}\stackrel{{\scriptstyle\triangledown}}{{x_{k}}},\dots,x_{n})-\sum_{k=1}^{i}(x_{1},{\dots}\stackrel{{\scriptstyle\triangledown}}{{x_{k}}},\dots,x_{n})=\\ (x_{1},\dots,\stackrel{{\scriptstyle\triangledown}}{{x_{i+1}}},\dots,x_{n}).\end{array}$$

$$sf(x_1,\dots ,x_i\stackrel{▽}{,}{x}_{i+1},\dots ,x_n)=s(x_1,\dots ,x_n)=(\stackrel{▽}{,}{x}_1,\dots ,x_n).$$

$$\begin{array}[]{llll}gr(x_{1},\dots,x_{i}\stackrel{{\scriptstyle\triangledown}}{{,}}x_{i+1},\dots,x_{n})=g\left(\sum_{k=1}^{i}(x_{1},\dots,\stackrel{{\scriptstyle\triangledown}}{{x_{k}}},\dots,x_{n})\right)=\\ \sum_{k=1}^{i}\left((x_{1},\dots,x_{k}\stackrel{{\scriptstyle\triangledown}}{{,}}\dots,x_{n})-(x_{1},{\dots}\stackrel{{\scriptstyle\triangledown}}{{,}}x_{k},\dots,x_{n})\right)=\\ -(\stackrel{{\scriptstyle\triangledown}}{{,}}\!x_{1},\dots,x_{n})+(x_{1},\dots,x_{i}\stackrel{{\scriptstyle\triangledown}}{{,}}x_{i+1},\dots,x_{n}).\end{array}$$

$$A_{\setminus D}=k(Q_{\setminus D})/⟨R{⟩}_{k(Q_{\setminus D})}$$

$${\mathsf{HH}}_{\ast }(A_{\setminus D})\simeq {\mathsf{HH}}_{\ast }(A).$$

$$B_F=k{Q}_F/⟨I{⟩}_{k{Q}_F}.$$

$$N=\underset{a\in F}{⨁}Bt(a)\otimes s(a)B.$$

$${\mathsf{HH}}_{\ast }(B_F)\simeq {\mathsf{HH}}_{\ast }(B).$$

$${\mathsf{dim}}_k{\mathsf{HH}}^{\ast }(B)={\mathsf{dim}}_k{\mathsf{HH}}^{\ast }(A)-{\mathsf{dim}}_k{\mathsf{Ext}}_A^{\ast }((fA{)}^{\prime },Ae).$$

$${\mathsf{dim}}_k\gamma =\prod _{i=1}^{n-1}{\mathsf{dim}}_{k}s(a_{i+1})Bt(a_i).$$

Full text

2010 MSC: 18G25, 16E40, 16E30, 18G15

Keywords: Hochschild, cohomology, homology, relative, quiver, arrow

1 Introduction

Hochschild (co)homology has been widely studied for algebras presented by quivers and relations, that is for bound quiver algebras. The main purpose of this work is to make precise how the Hochschild (co)homology changes when deleting or adding arrows.

In 1956 G. Hochschild [24] introduced a relative (co)homological theory, which amounts to consider an exact category, see [29, 7]. This theory has been used rarely, in part because of a lack of relation with the usual (co)homological theory given in [23]. Nevertheless, for an extension of algebras $B\subset A$ such that $A/B$ is a projective $B$-bimodule, A. Kaygun has established in 2012 [25, 26] a Jacobi-Zariski long exact sequence connecting both theories. Note that M. Auslander and Ø. Solberg have considered relative homological algebra, see [4] and [33].

In Section 2 we provide an account of the relative (co)homological theory and we show that for a tensor algebra there is a relative projective resolution of length one.

An inert arrow of a bound quiver algebra $A$ is an arrow of the quiver which is not involved in a minimal set of relations of $A$. Our main result in Section 3 is that deleting a set of inert arrows does not change the Hochschild homology in degrees greater or equal than $2$.

On the other hand in Section 4 we provide formulas for Hochschild cohomology, also when deleting inert arrows of a bound quiver algebra. For degrees greater or equal than $2$, the change of dimension is expressed in terms of the dimensions of $\mathsf{Ext}$ vector spaces between indecomposable injective and projective modules, with multiplicities given by the dimensions of some relative paths that we introduce in Section 4.

The proofs rely on the reverse procedure, that is on adding arrows. This way we obtain a tensor algebra over the original algebra for a suitable projective bimodule. This algebra is finite dimensional if and only if adding arrows does not create relative cycles. In case of adding just one arrow, which does not leads to a relative cycle, the tensor algebra is a trivial extension which has been already considered in [17]. In this paper E.L. Green, C. Psaroudakis and Ø. Solberg study this operation in relation to the finitistic dimension conjecture.

In Section 5 we apply our results to the amalgamation of a bound quiver algebra with a quiver without oriented cycles. In this section we also consider Hochschild cohomology of Gorenstein algebras, by adding arrows to its quiver.

The results that we obtain in this paper are in the framework of replacing an algebra by a closely related algebra, towards producing algorithmic ways for computing Hochschild (co)homology. Other work in this direction can be found for instance in [5, 6, 9, 10, 15, 16, 18, 22, 27, 28]. Hochschild (co)homology has been also considered for bound quiver algebras in relation to their representations and their deformations, see for instance [19, 21, 22, 32].

Hochschild (co)homology is a derived invariant, see for instance [30]. M. Gerstenhaber has shown in [14] that Hochschild (co)homology affords additional structure: cup and cap products, and the Gerstenhaber bracket. Together with Connes’ differential this constitutes the differential calculus, or Tamarkin-Tsygan calculus of an algebra, see for instance [34, 11]. This theory is also a derived invariant, see [1, 2]. It should be interesting to describe the effect of deleting an inert arrow of a bound quiver algebra on its differential calculus.

2 Relative Hochschild (co)homology and tensor algebras

Let $k$ be a field and let $B\subset A$ be an extension of $k$-algebras. Modules are left modules.

An $A$-module $P$ is relative projective if for any $A$-morphism $f:X\to Y$ with a $B$-section, and any $A$-morphism $g:P\to Y$, there exists an $A$-morphism $g^{\prime}:P\to X$ such that $fg^{\prime}=g$. The category of $A$-modules is an exact category with respect to the short exact sequences which are $B$-split, see [7]. As mentioned in [7], these notions are commonly attributed to D. Quillen [29]. The projective objects of this exact category (see [7, Definition 11.1]) are the relative projective modules.

An $A$-module is induced if it is isomorphic to $A\otimes_{B}U$, where $U$ is a $B$-module. As proved by Hochschild in [24], a module is relative projective if and only if it is isomorphic to a direct summand of an induced module.

Definition 2.1

[24]** A relative projective resolution $P_{\bullet}\to M$ of an $A$-module $M$ is a complex

[TABLE]

where $P_{i}$ is relative projective for all $i$ and there exists a $B$-contracting homotopy, that is there exist $B$-maps $t$ of degree $-1$ such that $dt+td=1$.

It can be easily shown that relative projective resolutions are the projective resolutions in the exact category with respect to $B$-split short exact sequences, see [7, Definition 12.1]. The usual comparison theorem holds for relative projective resolutions, see [24, p. 250] and [7, Theorem 12.4]. Hence the following definitions do not depend, up to isomorphism, on the choice of a relative projective resolution.

Definition 2.2

Let $M$ and $N$ be $A$-modules and let $P_{\bullet}\to M$ be a relative projective resolution. The cohomology of the cochain complex

[TABLE]

is $\mathsf{Ext}^{*}_{A|B}(M,N)$.

Let $M$ be a left $A$-module and let $N$ be a right $A$-module. Let $P_{\bullet}\to M$ be a relative projective resolution. The homology of the chain complex

[TABLE]

is $\mathsf{Tor}_{*}^{A|B}(N,M)$.

Next we recall the relative theory provided by Hochschild in [24] for bimodules.

Let $C$ and $D$ be $k$-algebras. The category of $C\!-\!D$-bimodules is identified with the category of $C\otimes D^{{\mathsf{o}p}}$-modules. A projective $C-D$-bimodule is a projective $C\otimes D^{\mathsf{o}p}$-module.

Definition 2.3

Let $M$ be an $A$-bimodule. The relative Hochschild cohomology and homology of $A$ respect to $B$ with coefficients in $M$ are respectively the graded vector spaces

[TABLE]

[TABLE]

There is a relative bar resolution of $A$ which provides the subsequent way of computing relative Hochschild (co)homology, see [24]. As mentioned in the Introduction, Kaygun obtained long exact sequences relating the usual Hochschild (co)homology with the relative one, see [25, 26].

Let $B$ be a $k$-algebra and let $N$ be a $B$-bimodule. The *tensor algebra *

[TABLE]

is a non negatively graded algebra with $T_{0}=B$ and $T_{n}=N^{\otimes_{B}n}$ for $n>0$. Its positive part is $T^{>0}=\oplus_{n>0}N^{\otimes_{B}n}$.

Example 2.4

Let $Q$ be a finite quiver, with set of vertices $Q_{0}$, set of arrows $Q_{1}$, and $s,t:Q_{1}\to Q_{0}$ the maps which associate to each arrow respectively its source and target vertices. The path algebra $kQ$ is the tensor algebra over the semisimple commutative algebra $kQ_{0}$ of the $kQ_{0}$-bimodule $kQ_{1}$. We denote by $Q_{n}$ the set of paths of length $n$, note that $(kQ)_{n}=kQ_{n}$.

A bound quiver algebra is an algebra $kQ/I$, where $I$ is an admissible ideal, that is $\langle Q_{n}\rangle\subset I\subset\langle Q_{2}\rangle$ for some $n$.

Let $k$ be an algebraically closed field. By results of P. Gabriel in [12, 13], see also for instance [3, Theorem 3.7] or [31], a finite dimensional $k$-algebra $A$ is Morita equivalent to a bound quiver algebra.

We recall the universal property of a tensor algebra $T=T_{B}(N)$. Let $\Lambda$ be a $k$-algebra. An algebra morphism $\varphi:T\to\Lambda$ is uniquely determined by an algebra morphism $\varphi_{0}:B\to\Lambda$ - which turns $\Lambda$ into a $B$-bimodule - and a $B$-bimodule morphism $\varphi_{1}:N\to\Lambda$.

Theorem 2.5

Let $T=T_{B}(N)$ be the tensor algebra over a $k$-algebra $B$ of a $B$-bimodule $N$. Let $X$ be a $T$-bimodule. For $*\geq 2$,

[TABLE]

[TABLE]

Proof. We will show that the following is a $(T\!-\!T|B\!-B)$ relative projective resolution of $T$:

[TABLE]

where $f$ is the product of $T$ and $g(x\otimes n\otimes y)=xn\otimes y-x\otimes ny.$ We obtain the results of the statement by using this relative resolution to compute Hochschild (co)homology.

It is straightforward to verify that $fg=0$. Clearly the involved $T$-bimodules are induced, hence they are relative projective.

In order to define a $(B\!-\!B)$-contracting homotopy, we introduce notation. Firstly, instead of $x_{1}\otimes\dots\otimes x_{n}$ we will write $x_{1},\dots,x_{n}$. Secondly observe that the tensor element $x_{1},\dots,x_{n}$ can be considered in different ways: for any decomposition $n=i+j$ with $i\geq 0$ and $j\geq 0$, it can be viewed as an element of

[TABLE]

In this case we write this element as $(x_{1},{\dots},x_{i}\stackrel{{\scriptstyle\triangledown}}{{,}}x_{i+1},\dots,x_{n}).$ For $n=0+n$ and $n=n+0$ we remark that the notation is:

[TABLE]

[TABLE]

Finally for $n=i+1+j$ with $i\geq 0$ and $j\geq 0$, the tensor $x_{1},\dots,x_{n}$ can also be considered as an element of

[TABLE]

In this case we denote it by $x_{1},\dots,x_{i},\stackrel{{\scriptstyle\triangledown}}{{x_{i+1}}},x_{i+2},\dots,x_{n}$. For $n=0+1+(n-1)$ and $n=(n-1)+1+0$ we write:

[TABLE]

[TABLE]

With these notations the morphisms $f$ and $g$ are as follows:

[TABLE]

[TABLE]

The sequence (2.1) is graded, it is the direct sum of the following sequences of $B$-bimodules:

[TABLE]

Next we define $s:T_{n}\to(T\otimes_{B}T)_{n}$ and $r:(T\otimes_{B}T)_{n}\to(T\otimes_{B}N\otimes_{B}T)_{n}$ and we prove that these maps verify the following:

• •

$r$ and $s$ are $B\!-\!B$-bimodule maps,

• •

$fs=1$ and $rg=1$,

• •

$gr+sf=1$

providing this way the required contracting homotopy, see Definition 2.1.

The morphism $s$ is defined by $s(x_{1},\dots,x_{n})=(\stackrel{{\scriptstyle\triangledown}}{{,}}\!x_{1},\dots,x_{n})$, note that it is well defined. Moreover $s$ is easily seen to be a $B\!-\!B$-section of $f$.

In order to define $r$, let $i\geq 0$, $j\geq 0$ with $i+j=n$. Let

[TABLE]

be given by

[TABLE]

Observe that $r_{0}=0$. Each $r_{i}$ is a well defined $B\!-\!B$-morphism. We put $r=\oplus_{i=0}^{n}r_{i}$. Next we verify that $rg=1$.

[TABLE]

Finally we verify that $gr+sf=1$.

[TABLE]

[TABLE]

In particular $(gr+sf)(\stackrel{{\scriptstyle\triangledown}}{{,}}\!x_{1},\dots,x_{n})=0+(\stackrel{{\scriptstyle\triangledown}}{{,}}\!x_{1},\dots,x_{n}).$ $\diamond$

3 Homology

Let $A=kQ/I$ be a bound quiver algebra, and let $R$ be a minimal finite subset of the path algebra $kQ$ such that $I=\langle R\rangle$. Note that elements $r\in I$ are possibly linear combination of parallel paths, that is of paths with the same source and the same target vertices, see for instance [3].

Definition 3.1

Let $A=kQ/\langle R\rangle$ be a bound quiver algebra. An arrow $a$ of $Q$ is inert if $a$ does not appear in any of the paths which provide the elements of $R$.

Let $D$ be a set of inert arrows, let $Q_{{\setminus}D}$ be the quiver $Q$ where the arrows of $D$ are deleted. We set

[TABLE]

where the denominator is the two sided ideal generated by $R$ in $k(Q_{{\setminus}D})$. Note that $A_{{\setminus}D}$ is a bound quiver algebra. Moreover $A_{{\setminus}D}$ is a subalgebra of $A$.

Theorem 3.2

Let $A=kQ/\langle R\rangle$ be a bound quiver algebra and let $D$ be a set of inert arrows of $Q$. For $*\geq 2$

[TABLE]

The proof will rely on the reverse procedure, namely adding new arrows.

Definition 3.3

Let $Q$ be a quiver and let $F$ be a finite set of new arrows, that is $F$ is a finite set with maps $s,t:F\to Q_{0}$. The new quiver $Q_{F}$ is given by $(Q_{F})_{0}=Q_{0}$ and $(Q_{F})_{1}=Q_{1}\cup F$, where the source and target maps are provided by the corresponding maps of $Q_{1}$ and $F$.

Let $B=kQ/I$ be a bound quiver algebra, and let $F$ be a set of new arrows. We set

[TABLE]

Remark 3.4

Suppose that $B_{F}$ is finite dimensional. Obviously the set of arrows $F$ is inert and $(B_{F})_{\setminus F}=B$.

If $A=kQ/\langle R\rangle$ is a bound quiver algebra and $D$ is a set of inert arrows, then $\left(A_{\setminus D}\right)_{D}=A$.

Given a bound quiver algebra $B$ and a set of new arrows $F$, consider the projective $B$-bimodule

[TABLE]

The following result is clear by using the universal property of tensor algebras.

Theorem 3.5

Let $B=kQ/\langle R\rangle$ be a bound quiver algebra, and let $F$ be a set of new arrows. The algebra $B_{F}$ is isomorphic to the tensor algebra $T=T_{B}(N)$.

Theorem 3.6

Let $B$ be a bound quiver algebra, and let $F$ be a set of new arrows. For $*\geq 2$

[TABLE]

Proof. By the previous result, $B_{F}\simeq T=T_{B}(N)$. The $B$-bimodule $T/B=T^{>0}$ is projective, hence the Jacobi-Zariski long exact sequence for Hochschild homology obtained by Kaygun holds, see [25, 26]. On the other hand, by Theorem 2.5 we have that $\mathsf{H}_{*}(T|B,T)=0$ for $*\geq 2$. Then $\mathsf{HH}_{*}(T)\simeq\mathsf{H}_{*}(B,T)$ for $*\geq 2$. Moreover $\mathsf{H}_{*}(B,T)=\mathsf{HH}_{*}(B)\oplus\mathsf{H}_{*}(B,T^{>0})$. The second summand is zero since the $B$-bimodule of coefficients is projective. $\diamond$

4 Cohomology

In case of deleting one inert arrow we will obtain the following result. Note that for $X$ a right module, we denote by $X^{\prime}$ the left module $\mathsf{Hom}_{k}(X,k)$.

Theorem 4.1

Let $A$ be a bound quiver algebra, let $a$ be an inert arrow from $e$ to $f$ and let $B=A_{\setminus\{a\}}$. For $*\geq 2$ we have

[TABLE]

See Remark 4.3 for a proof of the previous result.

Next we state a result for a set of deleted arrows. As in the previous section, the proof will be by the reverse procedure of adding arrows. However the Jacobi-Zariski long exact sequence for cohomology requires that the bimodule of coefficients is finite dimensional. To ensure this, and to state the result, we introduce the following.

Let $A=kQ/\langle R\rangle$ be a bound quiver algebra, let $D$ be a set of inert arrows and let $B=A_{{\setminus}D}$, see (3.1).

A pair of arrows $(a_{2},a_{1})$ of $D$ is linked if $s(a_{2})Bt(a_{1})\neq 0$.

• -

A relative $n$-path is a sequence $\gamma=a_{n}a_{n-1}\dots a_{2}a_{1}$ of arrows of $D$ such that $(a_{i+1},a_{i})$ is linked for $i=1,\dots,n-1$. The source $s(\gamma)$ and the target $t(\gamma)$ of $\gamma$ are respectively $s(a_{1})$ and $t(a_{n})$.

• -

The set of relative $n$-paths is denoted $\Gamma_{n}$, while the set of all relative paths is denoted $\Gamma$.

• -

The relative path $\gamma$ is a relative cycle if $(a_{1},a_{n})$ is linked. This way an arrow $a$ is a relative cycle if $(a,a)$ is linked, that is if $s(a)Bt(a)\neq 0$.

• -

We set

[TABLE]

If $\gamma=a\in F$, then $\mathsf{dim}_{k}\gamma=1$.

We will prove the following as a consequence of Theorem 4.6.

Theorem 4.2

Let $A$ be a bound quiver algebra, let $D$ be a set of inert arrows and let $B=A_{\setminus D}$. For $*\geq 2$ we have

[TABLE]

Remark 4.3

The proof of Theorem 4.1 follows: if $D=\{a\}$, then there is only one relative path $\gamma=a$. The formula above gives

[TABLE]

Moreover, at the end of this section we will prove that

[TABLE]

We first prove the following.

Proposition 4.4

Let $B$ be a bound quiver algebra, let $F$ be a set of new arrows and let $N$ be the associated projective $B$-bimodule, see (3.2). We have

[TABLE]

The algebra $T_{B}(N)=B_{F}$ is finite dimensional if and only if the length of the relative paths is bounded, that is there are no relative cycles.

Proof. We sketch the proof in case $F=\{a_{1},a_{2}\}$, where $a_{1}$ is from $e$ to $f$ and $a_{2}$ from $g$ to $h$, hence $N=(Bf\otimes eB)\oplus(Bh\otimes gB)$. One of the four direct summands of $N\otimes_{B}N$ is

[TABLE]

which is isomorphic as a $B$-bimodule to $\mathsf{dim}_{k}(gBf)Bh\otimes eB.$ If $(a_{2},a_{1})$ is linked, then this direct summand is non zero and it corresponds to the relative path $a_{2}a_{1}$ in the formula. The rest of the proof is along the same lines.

Lemma 4.5

Let $B$ be a bound quiver algebra, and let $e,f\in Q_{0}$. We have

[TABLE]

Proof. First we recall that if $U$ and $W$ are left $B$-modules, then

[TABLE]

see for instance [8, p. 170, 4.4]).

Let $U$ and $V$ be respectively left and right finite dimensional $B$-modules, so that the $B$-bimodules $U\otimes V$ and $\mathsf{Hom}_{k}(V^{\prime},U)$ are isomorphic. Then

[TABLE]

$\diamond$

Theorem 4.6

Let $B=kQ/\langle R\rangle$ be a bound quiver algebra, let $F$ be a set of new arrows and let $A=B_{F}=T_{B}(N)$. Suppose that there are no relative cycles, that is $A$ is finite dimensional. For $*\geq 2$

[TABLE]

Proof. We infer from the Jacobi-Zariski long exact sequence obtained by Kaygun in [25, 26] and from Theorem 2.5 that for $*\geq 2$ we have $\mathsf{HH}^{*}(A)\simeq\mathsf{H}^{*}(B,A).$ Moreover $\mathsf{H}^{*}(B,A)\simeq\mathsf{HH}^{*}(B)\oplus\mathsf{H}^{*}(B,T^{>0}).$ By Proposition 4.4

[TABLE]

Lemma 4.5 provides the result. $\diamond$

Theorem 4.2 is inferred from the previous result by adding as new arrows the deleted ones.

Lemma 4.7

Let $B$ be a bound quiver algebra. Let $a$ be a new arrow from $e$ to $f$ which does not provides a relative oriented cycle, and let $A=B_{F}=T_{B}(N)$. For $*\geq 2$

[TABLE]

Proof. The Kaygun’s Jacobi-Zariski long exact sequence and Theorem 2.5, show that for $*\geq 2$

[TABLE]

We have that $N\otimes_{B}N=0$ because $a$ is not a relative oriented cycle, that is $eBf=0$. Hence $A=B\oplus N$, where $N=Bf\otimes eB$. A simple computation shows that $Af\otimes eA=Bf\otimes eB$. Hence

[TABLE]

Lemma 4.5 provides the result. $\diamond$

We record the following result as a consequence of Theorem 4.1.

Corollary 4.8

Let $A$ be a bound quiver algebra, let $a$ be an inert arrow from $e$ to $f$ and let $B=A_{\setminus\{a\}}$. If $Ae$ is an injective module, and/or if $(fA)^{\prime}$ is a projective module, we have for $*\geq 2$

[TABLE]

In order to use the previous result, we recall the well known criterion for a projective indecomposable module to be injective.

Proposition 4.9

Let $A=kQ/I$ be a bound quiver algebra and let $e\in Q_{0}$. The indecomposable projective left module $Ae$ is injective if and only if $\mathsf{soc}Ae$ is a simple module $S_{y}$ for a vertex $y$, and $\mathsf{dim}_{k}(Ae)=\mathsf{dim}_{k}(yA)^{\prime}$. In that case $Ae\simeq(yA)^{\prime}$.

5 Examples

5.1 Amalgamation with a quiver without oriented cycles

Let $\Lambda=kQ/I$ be a bound quiver algebra, and let $S$ be a finite quiver without oriented cycles. We will construct a quiver where some vertices of $S$ and $Q$ are identified.

Let $Y\subset S_{0}$ and $X\subset Q_{0}$, with a bijective map $u:X\to Y$. Let $\sim$ be the equivalence relation on $X\cup Y$ given by $x\sim u(x)$. We extend it to an equivalence relation on $Q_{0}\cup S_{0}$ in the obvious manner, each other vertex is just equivalent to itself.

Definition 5.1

The amalgamated quiver $Q\cup_{u}S$ is given by

• •

$Q_{0}\cup_{u}S_{0}/\sim$,

• •

$(Q\cup_{u}S)_{1}=Q_{1}\cup S_{1}$.

The amalgamated algebra $\Lambda\cup_{u}S$ is $k(Q\cup_{u}S)/\langle I\rangle_{k(Q\cup_{u}S)}$.

Theorem 5.2

Let $\Lambda=kQ/I$ be a bound quiver algebra, let $S$ be a finite quiver without oriented cycles and let $u$ be as above. Suppose that for each path of positive length of $S$, one of its extremities is not in $Y$. For $*\geq 2$

[TABLE]

Proof. Let $G$ be the quiver $Q$ with the set $S_{0}\setminus Y$ of new vertices added - each one is a connected component of $G$. Consider the algebra B=\Lambda\times\mbox{{\Large\times}}_{|S_{0}\setminus Y|}k. Hochschild cohomology is additive on the product of algebras, and $\mathsf{HH}^{*}(k)=0$ for $*\geq 1$, hence $\mathsf{HH}^{*}(\Lambda)=\mathsf{HH}^{*}(B)$ for $*\geq 1$.

Recall that $G_{S_{1}}$ is the quiver $G$ with the set $S_{1}$ of new arrows added, we have $G_{S_{1}}=Q\cup_{u}S$. The hypotheses on the paths of $S$ imply that there are no relative oriented cycles, hence $B_{S_{1}}=\Lambda\cup_{u}S$ is finite dimensional. Moreover, for each relative path at least one of its extremities is an isolated vertex $x\in S_{0}\setminus Y$. We have that $(xB)^{\prime}=Bx$, which is a simple module. The corresponding $\mathsf{Ext}_{B}$’s in Theorem 4.6 vanish. $\diamond$

5.2 Gorenstein algebras

A finite dimensional $k$-algebra $B$ is Gorenstein if $B$ is of finite injective dimension $n_{1}$ as a left module, and the injective left module $B^{\prime}$ is of finite projective dimension $n_{2}$, see for instance [20]. The algebra is called $n$-Gorenstein for $n$ the maximum of these numbers.

Let $B=kQ/\langle R\rangle$ be a bound quiver algebra. If $B$ is $n$-Gorenstein, then for any $x\in Q_{0}$ the projective dimension of $(xB)^{\prime}$ and the injective dimension of $Bx$ is at most $n$. For instance selfinjective algebras are [math]-Gorenstein algebras. The following result is a consequence of Theorem 4.6.

Proposition 5.3

Let $B=kQ/I$ be a bound quiver algebra which is $n$-Gorenstein. Let $\widehat{Q}$ be the quiver obtained from $Q$ by adding a finite number of new arrows, and suppose $\widehat{B}=k\widehat{Q}/\langle I\rangle_{\widehat{Q}}$ is finite dimensional. For $n>0$ and $*\geq n+1$,

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while for $n=0$ - that is if $B$ is selfinjective- the isomorphism holds for $*\geq 2$.

Example 5.4

For $m\geq 2$, let $Q$ be the following cyclic connected quiver:

s_{0}$$s_{1}$$s_{i}$$s_{m-1}$$a_{0}$$a_{1}$$a_{m-2}$$a_{m-1}$$a_{i}

For $2\leq l<m$, let $B=kQ/\langle Q_{l}\rangle$, which is a selfinjective algebra. Let $\widehat{Q}$ be the quiver obtained from $Q$ by adding a finite number of arrows at pairs of vertices. Suppose that $\widehat{B}=k\widehat{Q}/\langle Q_{l}\rangle_{\widehat{Q}}$ is finite dimensional, that is there are no relative oriented cycles. For $*\geq 2$

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