Generalized parallel paths method for computing the first Hochschild cohomology groups with applications to Brauer graph algebras

Let $A=kQ/I$ be a quiver algebra where $I$ is generated by a set of relations. In this subsection we recall from [Green] the Gröbner basis theory for the ideal $I$. Let us first introduce a special kind of well-order on the basis $Q_{\geq 0}$ of the path algebra $kQ$. By [Green, Section 2.2.2], a well-order $>$ on $Q_{\geq 0}$ is called admissible if it satisfies the following conditions where $p,q,r,s\in Q_{\geq 0}$:

• •

  if $p<q$ then $pr<qr$ if both $pr\neq 0$ and $qr\neq 0$.

• •

  if $p<q$ then $sp<sq$ if both $sp\neq 0$ and $sq\neq 0$.

• •

  if $p=qr$, then $p\geq q$ and $p\geq r$.

Given a quiver $Q$ as above, there are “natural” admissible orders on $Q_{\geq 0}$. Here is one example:

We now fix an admissible well-order $>$ on $Q_{\geq 0}$. For any $a\in kQ$, we have $a=\sum_{p\in Q_{\geq 0},\;\lambda_{p}\in k}\lambda_{p}p$ and write $\mathrm{Supp}(a)=\{p\mid\lambda_{p}\neq 0\}$. We call $\mathrm{Tip}(a)=p$, if $p\in\mathrm{Supp}(a)$ and $p^{\prime}\leq p$ for all $p^{\prime}\in\mathrm{Supp}(a)$. Then we denote the tip of a set $W\subseteq kQ$ by $\mathrm{Tip}(W)=\{\mathrm{Tip}(w)|w\in W\}$ and write $\mathrm{NonTip}(W):=Q_{\geq 0}\backslash\mathrm{Tip}(W)$. We also denote the coefficient of the tip of $a$ by $\mathrm{CTip}(a)$. In particular, we will use $\mathrm{Tip}(I)$ and $\mathrm{NonTip}(I)$ for the ideal $I$ of $kQ$. By [Green], there is a decomposition of vector spaces

$$kQ=I\oplus\mathrm{Span}_{k}(\mathrm{NonTip}(I)).$$

So $\mathrm{NonTip}(I)$ (modulo $I$) gives a “monomial” $k$-basis of the quotient algebra $A=kQ/I$.

Actually, in this case $I=\langle\mathcal{G}\rangle$. By the discussion in [Green], we have a complete method to judge whether a set of generators of an ideal $I$ in $kQ$ is a Gröbner basis, which is called the Termination Theorem. The idea is to check whether some special elements of the ideal $I$ are divisible by this basis, instead of to check all the elements in $I$.

It is easy to see that under a given admissible order, $I$ has a unique reduced Gröbner basis $\mathcal{G}$, and in this case $\mathrm{Tip}(\mathcal{G})$ is a minimal generator set of $\langle\mathrm{Tip}(I)\rangle$; moreover, $b\in Q_{\geq 0}$ lies in $\mathrm{NonTip}(I)$ if and only if $b$ is not divided by any element of $\mathrm{Tip}(\mathcal{G})$. In the following, we always assume that $\mathcal{G}$ is a reduced Gröbner basis of $I$.

Note that when $\mathcal{G}$ is reduced, there is a one-to-one correspondence between $\mathcal{G}$ and $\mathrm{Tip}(\mathcal{G})$: for $g\in\mathcal{G}$, $\mathrm{Tip}(g)\in\mathrm{Tip}(\mathcal{G})$; conversely, for $w\in\mathrm{Tip}(\mathcal{G})$, there is a unique $g\in\mathcal{G}$ such that $w=\mathrm{Tip}(g)$. We shall denote the correspondence from $\mathcal{G}$ to $\mathrm{Tip}(\mathcal{G})$ by $\mathrm{Tip}$ and its inverse by $\mathrm{Tip^{-1}}$.

Now let $E:\simeq\oplus_{e\in Q_{0}}ke$ be the separable subalgebra of $A$ generated by the classes modulo $I$ of the vertices of $Q$, such that $A=E\oplus A_{+}$ as $E^{e}$-modules, where $E^{e}=E\otimes_{k}E$ and $A_{+}:=\mathrm{Span}_{k}\{\mathrm{NonTip}(I)\backslash Q_{0}\}$. Actually, there is an $E^{e}$-projection from $A$ to $A_{+}$, denoted by $p_{A}$. The reduced bar resolution of the quiver algebra $A$ can be written by the form of the following theorem in the sense of Cibils [C].

Since the reduced bar resolution is a two-sided projective resolution of $A$, we can use it to compute the Hochschild cohomology groups $\mathrm{HH}^{n}(A)$ of $A$. More concretely, applying the functor $\mathrm{Hom}_{{A}^{e}}(-,{A})$ to $B(A)$ we get a cochain complex $(C^{*}(A),\delta^{*})$, where $C^{0}(A)\cong\mathrm{Hom}_{E^{e}}(E,A)\cong A^{E}=\{a\in A\mid sa=as\text{ for all }s\in E\}$, $C^{n}(A)=\mathrm{Hom}_{A^{e}}(B_{n}(A),A)\cong\mathrm{Hom}_{E^{e}}((A_{+})^{\otimes_{E}n},A)$ for $n\geq 1$ (cf. Lemma 3.3), $(\delta^{0}a)(x)=ax-xa$ for $a\in A^{E}$ and $x\in A_{+}$, and

$$(\delta^{n}f)(x_{1}\otimes\cdots\otimes x_{n+1})=x_{1}f(x_{2}\otimes\cdots\otimes x_{n+1})$$

$$+\sum_{i=1}^{n}(-1)^{i}f(x_{1}\otimes\cdots\otimes x_{i}x_{i+1}\otimes\cdots\otimes x_{n+1})+(-1)^{n+1}f(x_{1}\otimes\cdots\otimes x_{n})x_{n+1}.$$

Then we have

$$\mathrm{HH}^{n}(A)=\mathrm{Ker}\delta^{n}/\mathrm{Im}\delta^{n-1}.$$

In particular, we have $\mathrm{HH}^{1}(A)=\mathrm{Ker}\delta^{1}/\mathrm{Im}\delta^{0}$ as $k$-spaces, where $\mathrm{Ker}\delta^{1}$ is the set of $E^{e}$-derivations of $A_{+}$ into $A$ and the elements in $\mathrm{Im}\delta^{0}$ are inner $E^{e}$-derivations of $A_{+}$ into $A$. Note that we can identify $\mathrm{Der}_{E^{e}}(A_{+},A)$ with $\mathrm{Der}_{E^{e}}(A,A)$ and $\mathrm{Ker}\delta^{1}=\mathrm{Der}_{E^{e}}(A_{+},A)$ has a Lie algebra structure under the Lie bracket

$$[f,g]_{HH}:=f\circ p_{A}\circ g-g\circ p_{A}\circ f$$

for $f,g\in\mathrm{Der}_{E^{e}}(A_{+},A)$, where $p_{A}$ denotes the $E^{e}$-projection from $A$ to $A_{+}$. Moreover, $\mathrm{Im}\delta^{0}$ is a Lie ideal of $\mathrm{Ker}\delta^{1}$, so that $\mathrm{HH}^{1}(A)$ is a Lie algebra. This structure was first defined by Gerstenhaber [Ger] using the standard bar resolution of $A$.

In next two subsections we will explain how to use the algebraic Morse theory to shrink the above reduced bar resolution of $A$ to a “smaller” one, such that the homology of the two complexes coincides.

The most general version of algebraic Morse theory was presented in Chen, Liu and Zhou [CLZ]. For our purpose, we will adopt to a Morse matching condition defined in [CLZ, Proposition 3.2].

Let $R$ be an associative ring and $C_{*}=(C_{n},\partial_{n})_{n\in\mathbb{Z}}$ be a chain complex of left $R$-modules. We assume that each $R$-module $C_{n}$ has a decomposition $C_{n}\simeq\oplus_{i\in I_{n}}C_{n,i}$ of $R$-modules, so we can regard the differentials $\partial_{n}$ as a matrix $\partial_{n}=(\partial_{n,ji})$ with $i\in I_{n}$ and $j\in I_{n-1}$ and where $\partial_{n,ji}:C_{n,i}\rightarrow C_{n-1,j}$ is a homomorphism of $R$-modules.

Given the complex $C_{*}$ as above, we construct a weighted quiver $G(C_{*}):=(V,E)$. The set $V$ of vertices of $G(C_{*})$ consists of the pairs $(n,i)$ with $n\in\mathbb{Z},i\in I_{n}$ and the set $E$ of weighted arrows is given by the rule: if the map $\partial_{n,ji}$ does not vanish, draw an arrow in E from $(n,i)$ to $(n-1,j)$ and denote the weight of this arrow by the map $\partial_{n,ji}$.

A full subquiver $\mathcal{M}$ of the weighted quiver $G(C_{*})$ is called a partial matching if it satisfies the following two conditions:

• •

  $(Matching)$ Each vertex in $V$ belongs to at most one arrow of $\mathcal{M}$.

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  $(Invertibility)$ Each arrow in $\mathcal{M}$ has its weight invertible as a $R$-homomorphism.

With respect to a partial matching $\mathcal{M}$, we can define a new weighted quiver $G_{\mathcal{M}}(C_{*})=(V,E_{\mathcal{M}})$, where $E_{\mathcal{M}}$ is given by

• •

  Keep everything for all arrows which are not in $\mathcal{M}$ and call them thick arrows.

• •

  For an arrow in $\mathcal{M}$, replace it by a new dotted arrow in the reverse direction and the weight of this new arrow is the negative inverse of the weight of original arrow.

A path in $G_{\mathcal{M}}(C_{*})$ is called a zigzag path if dotted arrows and thick arrows appear alternately.

Next, for convenience, we will introduce from Jöllenbeck and Welker [JW] the notations related to the weighted quiver $G(C_{*})=(V,E)$ with a partial matching $\mathcal{M}$ on it.

Following [CLZ, Proposition 3.2], we call a partial matching $\mathcal{M}$ as above a Morse matching if any zigzag path starting from $(n,i)$ is of finite length for each vertex $(n,i)$ in $G_{\mathcal{M}}(C_{*})$.

Now we can define a new complex $C_{*}^{\mathcal{M}}$, which we call the Morse complex of $C_{*}$ with respect to $\mathcal{M}$. The complex $C_{*}^{\mathcal{M}}=(C_{n}^{\mathcal{M}},\partial_{n}^{\mathcal{M}})_{n\in\mathbb{Z}}$ is defined by

$$C_{n}^{\mathcal{M}}:=\oplus_{(n,i)\in V_{n}^{\mathcal{M}}}C_{n,i},$$

$$\partial_{n}^{\mathcal{M}}:\left\{\begin{array}[]{*{3}{lll}}C_{n}^{\mathcal{M}}&\rightarrow&C_{n-1}^{\mathcal{M}}\\ x\in C_{n,i}&\mapsto&\sum_{(n-1,j)\in V_{n-1}^{\mathcal{M}}}\Gamma((n,i),(n-1,j))(x).\end{array}\right.$$

The main theorem of algebraic Morse theory can be stated as follows.

Starting from the reduced bar resolution of an one-vertex quiver algebra $A$ which is viewed as a chain complex of projective $A^{e}$-modules, Sköldberg [ES] constructed a “smaller” $A^{e}$-projective resolution of $A$ using algebraic Morse theory, which is called the two-sided Anick resolution of $A$. It was pointed out in [CLZ] that Sköldberg’s construction generalizes to general quiver algebras.

Let $A=kQ/I$ be a quiver algebra, let $\mathcal{G}$ be a reduced Gröbner basis of the ideal $I$, and denote $W:=\mathrm{Tip}(\mathcal{G})$. Denote by $B(A)=(B_{*}(A),d_{*})$ the reduced bar resolution of $A$ (cf. Section 2.2). Similar as in [CLZ], we define a new quiver $Q_{W}=(V,E)$ with respect to $W$, which is called the Ufnarovskiĭ graph (or just Uf-graph).

Using Uf-graph $Q_{W}$ one can define (for each $i\geq-1$) the $i$-chains, which form a subset of generators of $B_{i}(A)=\bigoplus A^{e}[w_{1}|\cdots|w_{i}]$ for $i\geq 0$, with $w_{1},\cdots,w_{i}\in\mathrm{NonTip}(I)\backslash Q_{0}$ and $w_{1}\cdots w_{i}\in Q_{\geq 0}$.

By using the definition above, we can define a partial matching $\mathcal{M}$ to be the set of arrows of the following form in the weighted quiver $G(B_{*})$, where $B_{*}=B(A)$ is the reduced bar resolution of the algebra $A=kQ/I$:

$$[w_{1}|\cdots|w_{i+1}|w_{i+2}^{\prime}|w_{i+2}^{\prime\prime}|w_{i+3}|\cdots|w_{n}]\stackrel{{\scriptstyle(-1)^{i+2}}}{{\longrightarrow}}[w_{1}|\cdots|w_{i+2}|\cdots|w_{n}]$$

where

$$w=w_{1}\cdots w_{n}=w_{1}\cdots w_{i+2}^{\prime}w_{i+2}^{\prime\prime}\cdots w_{n},~{}~{}w_{i+2}=w_{i+2}^{\prime}w_{i+2}^{\prime\prime},$$

$$[w_{1}|\cdots|w_{i+1}|w_{i+2}^{\prime}|w_{i+2}^{\prime\prime}|w_{i+3}|\cdots|w_{n}]\in V_{w,i+1}^{(n+1)},~{}~{}[w_{1}|\cdots|w_{i+2}|\cdots|w_{n}]\in V_{w,i}^{(n)}.$$

Therefore, by Theorem 2.12 and Theorem 2.9, $(B^{\mathcal{M}}(A),d^{\mathcal{M}})$ is also a $A^{e}$-projective resolution of $A$, but the projective $A^{e}$-modules in $B^{\mathcal{M}}(A)$ are usually much smaller than $B(A)$. The resolution $(B^{\mathcal{M}}(A),d^{\mathcal{M}})$ is called the two-sided Anick resolution of $A$ in [ES] and [CLZ].

Note that for $n\geq 0$, the $n$-th component of the two-sided Anick resolution $B^{\mathcal{M}}(A)$ is $A\otimes_{E}kW^{(n-1)}\otimes_{E}A$. In particular, we have the following identifications:

$$W^{(-1)}=Q_{0},~{}~{}W^{(0)}=\{[w_{1}]\ |\ w_{1}\in Q_{1}\}\cong Q_{1},$$

$$W^{(1)}=\{[w_{1}|w_{2}]\ |\ \;w_{1}\in Q_{1},\;w_{1}w_{2}\in\mathrm{Tip}(\mathcal{G})\}\cong\mathrm{Tip}(\mathcal{G})=W.$$

Let ${A}=kQ/I$ be a quiver algebra and $E\simeq kQ_{0}$ be its separable subalgebra as in Section 2.2. Remind that we assume $I\subseteq kQ_{\geq 2}$. The following notations (most of them are taken from [Strametz]) are useful.

We now give a brief review about Strametz’s method in [Strametz] for computing the first Hochschild cohomology groups of monomial algebras. Recall that ${A}=kQ/I$ is a monomial algebra means that the ideal $I$ is generated by a set $Z$ of paths in $Q$. We shall assume that $Z$ is minimal, so that $Z$ is a reduced Gröbner basis of $I$ with $Z=\mathrm{Tip}(Z)$. Note that $\mathrm{NonTip}(Z)$ modulo $I$ gives a $k$-basis of $A$, which we denote by $\mathcal{B}$.

The proof of Proposition 3.2 uses the minimal $A^{e}$-projective resolution of the monomial algebra $A$ given by Bardzell [Bardzell] and the following two general lemmas.