Algebraic structure on Tate-Hochschild cohomology of a Frobenius algebra

A chain complex $C=(C,d^{C})$ over an algebra $A$ is the pair of a graded $A$-module $C=\bigoplus_{n\in\mathbb{Z}}C_{n}$ and a graded $A$-linear map $d^{C}:C\rightarrow C$ of degree $-1$ such that $(d^{C})^{2}=0$. Dually, a cochain complex $C=(C,d_{C})$ is the pair of a graded $A$-module $C=\bigoplus_{n\in\mathbb{Z}}C^{n}$ and a graded $A$-linear map $d_{C}:C\rightarrow C$ of degree $1$ such that $d_{C}^{2}=0$. In both cases, the graded map $d$ is called the differential of $C$. Note that any cochain complex $C$ can be regarded as a chain complex by reindexing $C_{n}=C^{-n}$, and vice versa. For a chain complex $C$ and $i\in\mathbb{Z}$, we define the $i$-shifted chain complex $\boldsymbol{\Sigma}^{i}C=(\boldsymbol{\Sigma}^{i}C,d^{\boldsymbol{\Sigma}^{i}C})$

to be the chain complex given by $(\boldsymbol{\Sigma}^{i}C)_{n}:=C_{n-i}$ and $d^{\boldsymbol{\Sigma}^{i}C}:=(-1)^{i}d^{C}$. For two chain complexes $C$ and $C^{\prime}$ of $A$-modules, a chain map $f:C\rightarrow C^{\prime}$ is a graded $A$-linear map $f:C\rightarrow C^{\prime}$ of degree $0$ such that $d^{C^{\prime}}f=fd^{C}$.

For two chain maps $f,g:C\rightarrow C^{\prime}$, we say that $f$ is homotopic to $g$ if there exists a graded $A$-linear map $h:C\rightarrow C^{\prime}$ of degree $1$ such that $f-g=d^{C^{\prime}}h+hd^{C}$. Then the graded map $h$ is called a homotopy from $f$ to $g$. In such a case, we denote $f\sim g$. A chain map $f:C\rightarrow C^{\prime}$ is called null-homotopic if $f\sim 0$. It is well-known that the relation $\sim$ is an equivalence relation. We denote by $[C,C^{\prime}]$ the quotient $k$-module induced by this equivalence relation. A chain map $f:C\rightarrow C^{\prime}$ is a homotopy equivalence if there exists a chain map $g:C^{\prime}\rightarrow C$ such that $fg\sim\operatorname{id}\nolimits_{C^{\prime}}$ and $gf\sim\operatorname{id}\nolimits_{C}$, and a chain map $f:C\rightarrow C^{\prime}$ is a quasi-isomorphism if the morphism $\operatorname{H}\nolimits_{n}(f):\operatorname{H}\nolimits_{n}(C)\rightarrow\operatorname{H}\nolimits_{n}(C^{\prime})$ induced by $f$ is an isomorphism for any $n\in\operatorname{\mathbb{Z}}\nolimits$, where $\operatorname{H}\nolimits_{n}(C)$ is the $n$-th homology group of the complex $C$ defined by the quotient $A$-module $Z_{n}(C)/B_{n}(C)$ with $Z_{n}(C)=\operatorname{Ker}\nolimits d^{C}_{n}$ and $B_{n}(C)=\operatorname{Im}\nolimits d^{C}_{n+1}$. We say that a chain complex $C$ is acyclic if $\operatorname{H}\nolimits_{n}(C)=0$ for every $n\in\operatorname{\mathbb{Z}}\nolimits$ and that a chain complex $C$ is contractible if $id_{C}\sim 0$. Then a homotopy from $\operatorname{id}\nolimits_{C}$ to $0$ is called a contracting homotopy.

The following easy and well-known lemma and its dual lemma play important roles in this paper. We will include a proof of the first lemma.

Let $C$ and $C^{\prime}$ be two chain complexes of $A$-modules. We define the tensor product $C\otimes_{A}C^{\prime}$ as the chain complex with components

and differentials $d^{C\otimes_{A}C^{\prime}}_{n}:(C\otimes_{A}C^{\prime})_{n}\rightarrow(C\otimes_{A}C^{\prime})_{n-1}$ given by

for homogeneous elements $x\in C$ and $x^{\prime}\in C^{\prime}$.

Now we define $\operatorname{\operatorname{\mathscr{Hom}}\nolimits}_{A}(C,C^{\prime})$ to be the chain complex with components

and differentials

given by

for any $f=(f_{i})_{i\in\operatorname{\mathbb{Z}}\nolimits}\in\operatorname{\operatorname{\mathscr{Hom}}\nolimits}_{A}(C,C^{\prime})_{n}$. One sees that $\operatorname{H}\nolimits_{n}(\operatorname{\operatorname{\mathscr{Hom}}\nolimits}_{A}(C,C^{\prime}))=[C,\boldsymbol{\Sigma}^{-n}C^{\prime}]$ for any $n\in\operatorname{\mathbb{Z}}\nolimits$. We put $[C,C^{\prime}]_{n}:=[C,\boldsymbol{\Sigma}^{-n}C^{\prime}]$.

Let $C$ be a chain complex of left $A$-modules. We define the $k$-dual complex $\operatorname{\mathbb{D}}\nolimits(C)$ of $C$ by the cochain complex $\operatorname{\operatorname{\mathscr{Hom}}\nolimits}_{k}(C,k)$ of right $A$-modules. Moreover, the $A$-dual complex $C^{\vee}$ of $C$ is defined as the cochain complex $\operatorname{\operatorname{\mathscr{Hom}}\nolimits}_{A}(C,A)$ of right $A$-modules.

Let $C,C^{\prime}$ and $C^{\prime\prime}$ be chain complexes of $A$-modules. The composition of graded maps is defined to be the chain map

sending $u\otimes v\in\operatorname{\operatorname{\mathscr{Hom}}\nolimits}_{A}(C^{\prime},C^{\prime\prime})_{s}\otimes\operatorname{\operatorname{\mathscr{Hom}}\nolimits}_{A}(C,C^{\prime})_{r}$ to

Hence the chain map induces a well-defined operator

Let $C_{1},C_{2},C^{\prime}_{1}$ and $C^{\prime}_{2}$ be chain complexes of $A$-modules, and let $u:C_{1}\rightarrow C^{\prime}_{1}$ and $v:C_{2}\rightarrow C^{\prime}_{2}$ be graded maps of degree $s$ and of degree $t$, respectively. The tensor product $u\otimes_{A}v$ of $u$ and $v$ is defined as the graded map of degree $s+t$ given by

for homogeneous elements $x_{1}\in C_{1}$ and $x_{2}\in C_{2}$. The tensor product of graded maps gives rise to a chain map

Throughout this section and the next, let $A$ be a $k$-algebra which is projective over $k$. Let $A^{\textrm{e}}$ denote the enveloping algebra $A\otimes A^{\circ}$ of $A$, where $A^{\circ}$ is the opposite algebra of $A$. Suppose that all projective resolutions of $A$ are taken over $A^{\textrm{e}}$. We view any module $M$ as a complex concentrated in degree $0$. Let $\operatorname{\mathbf{P}}\nolimits=\bigoplus_{i\geq 0}P_{i}$ be a projective resolution of $A$, that is, a quasi-isomorphism $\operatorname{\mathbf{P}}\nolimits\xrightarrow{\varepsilon}A$ with $P_{i}$ finitely generated projective. The epimorphism $\varepsilon:P_{0}\rightarrow A$ is called the augmentation of $\operatorname{\mathbf{P}}\nolimits$.

The $n$-th Hochschild cohomology group $\operatorname{H}\nolimits^{n}(A,M)$ of $A$ with coefficients in an $A$-bimodule $M$ is defined by $\operatorname{H}\nolimits^{n}(A,M):=\operatorname{H}\nolimits^{n}(\operatorname{\operatorname{\mathscr{Hom}}\nolimits}_{A^{\textrm{e}}}(\operatorname{\mathbf{P}}\nolimits,M))$, where the $n$-th component of $\operatorname{\operatorname{\mathscr{Hom}}\nolimits}_{A^{\textrm{e}}}(\operatorname{\mathbf{P}}\nolimits,M)$ is given by

The $n$-th Hochschild homology group $\operatorname{H}\nolimits_{n}(A,M)$ of $A$ with coefficients in an $A$-bimodule $M$ is defined as $\operatorname{H}\nolimits_{n}(A,M):=\operatorname{H}\nolimits_{n}(\operatorname{\mathbf{P}}\nolimits\otimes_{A^{\textrm{e}}}M)$.

In this section, we recall the definitions of the cup product and of the cap product in Hochschild theory. If $\operatorname{\mathbf{P}}\nolimits\xrightarrow{\varepsilon}A$ is a projective resolution, we can associate $\operatorname{\mathbf{P}}\nolimits$ with the following augmented chain complex having $A$ in degree $-1$:

For two projective resolutions $\operatorname{\mathbf{P}}\nolimits\xrightarrow{\varepsilon}A$ and $\operatorname{\mathbf{P}}\nolimits^{\prime}\xrightarrow{\varepsilon^{\prime}}A$, a chain map $f:\operatorname{\mathbf{P}}\nolimits\rightarrow\operatorname{\mathbf{P}}\nolimits^{\prime}$ is called an augmentation-preserving chain map if $\varepsilon^{\prime}f_{0}=\varepsilon$. Since the tensor product $\operatorname{\mathbf{P}}\nolimits\otimes_{A}\operatorname{\mathbf{P}}\nolimits$ of $\operatorname{\mathbf{P}}\nolimits$ with itself is also a projective resolution of $A$ with augmentation $\varepsilon\otimes_{A}\varepsilon:P_{0}\otimes_{A}P_{0}\rightarrow A$, there exists an augmentation-preserving chain map $\Delta:\operatorname{\mathbf{P}}\nolimits\rightarrow\operatorname{\mathbf{P}}\nolimits\otimes_{A}\operatorname{\mathbf{P}}\nolimits$. We call such a chain map a diagonal approximation. For any $A$-bimodules $M$ and $N$, we define a graded $k$-linear map

by

for homogeneous elements $u\in\operatorname{\operatorname{\mathscr{Hom}}\nolimits}_{A^{\textrm{e}}}(\operatorname{\mathbf{P}}\nolimits,M)$ and $v\in\operatorname{\operatorname{\mathscr{Hom}}\nolimits}_{A^{\textrm{e}}}(\operatorname{\mathbf{P}}\nolimits,N)$. Since the diagonal approximation $\Delta$ is a chain map, we see that the map $\smile$ is a chain map. Thus it induces a well-defined operator

For $u\in\operatorname{H}\nolimits^{m}(A,M)$ and $v\in\operatorname{H}\nolimits^{n}(A,N)$, we call $u\smile v\in\operatorname{H}\nolimits^{m+n}(A,M\otimes_{A}N)$ the cup product of $u$ and $v$.

Consider the chain map

defined by $u\otimes v\mapsto(u\otimes_{A}\operatorname{id}\nolimits_{M})v$ for homogeneous elements $u\in\operatorname{\operatorname{\mathscr{Hom}}\nolimits}_{A^{\textrm{e}}}(\operatorname{\mathbf{P}}\nolimits,M)$ and $v\in\operatorname{\operatorname{\mathscr{Hom}}\nolimits}_{A^{\textrm{e}}}(\operatorname{\mathbf{P}}\nolimits,\operatorname{\mathbf{P}}\nolimits\otimes_{A}N)$. Then it induces a well-defined operator, called the composition product,

Since $A$ is projective as a right $A$-module, the augmented chain complex $\operatorname{\mathbf{P}}\nolimits\xrightarrow{\varepsilon}A$ is contractible as a complex of right $A$-modules, so that the tensor product $\operatorname{\mathbf{P}}\nolimits\otimes_{A}N\rightarrow A\otimes_{A}N$ is acyclic, which means that $\varepsilon\otimes_{A}\operatorname{id}\nolimits_{N}:\operatorname{\mathbf{P}}\nolimits\otimes_{A}N\rightarrow N$ is a quasi-isomorphism. It follows from [Bro94, Chapter I, Theorem 8.5] that $\varepsilon\otimes_{A}\operatorname{id}\nolimits_{N}$ induces an isomorphism

We now recall the definition of the cap product between Hochschild cohomology and homology groups and the statement analogue to Theorem 1.4. Consider the chain map

given by

where $u\in\operatorname{\operatorname{\mathscr{Hom}}\nolimits}_{A^{\textrm{e}}}(\operatorname{\mathbf{P}}\nolimits,M)$ and $n\otimes_{A^{\textrm{e}}}x\otimes_{A}y\in((\operatorname{\mathbf{P}}\nolimits\otimes_{A}\operatorname{\mathbf{P}}\nolimits)\otimes_{A^{\textrm{e}}}N)$ are homogeneous elements and we use an isomorphism

Then the chain map

defined to be the composition of two chain maps $\gamma$ and

gives rise to a well-defined operator

For $u\in\operatorname{H}\nolimits^{m}(A,M)$ and $w\in\operatorname{H}\nolimits_{n}(A,N)$, we call $u\frown w\in\operatorname{H}\nolimits_{n-m}(A,M\otimes_{A}N)$ the cap product of $u$ and $w$.

On the other hand, we define a chain map

by

for homogeneous elements $u\in\operatorname{\operatorname{\mathscr{Hom}}\nolimits}_{A^{\textrm{e}}}(\operatorname{\mathbf{P}}\nolimits,M)$ and $(x\otimes_{A}y)\otimes_{A^{\textrm{e}}}n\in(\operatorname{\mathbf{P}}\nolimits\otimes_{A}\operatorname{\mathbf{P}}\nolimits)\otimes_{A^{\textrm{e}}}N$. Moreover, it follows from [Bro94, Chapter I, Theorem 8.6] that there exists an isomorphism

One proves the following in a similar way to the proof of Theorem $\ref{theo:3}(2)$ below.

Let us begin with the preparation for some notation. We denote by $\operatorname{Aut}\nolimits(A)$ the group of algebra automorphisms of $A$. Note that any $\alpha\in\operatorname{Aut}\nolimits(A)$ gives rise to $\alpha^{\circ}\in\operatorname{Aut}\nolimits(A^{\circ})$ defined by $\alpha^{\circ}(a^{\circ}):=\alpha(a)^{\circ}$ for any $a^{\circ}\in A^{\circ}$. For an $A$-bimodule $M$ and two automorphisms $\alpha,\beta\in\operatorname{Aut}\nolimits(A)$, we denote by ${}_{\alpha}M_{\beta}$ the $A$-bimodule which is $M$ as a $k$-module and whose $A$-bimodule structure is given by $a*m*b:=\alpha(a)m\beta(b)$ for $a,b\in A$ and $m\in{}_{\alpha}M_{\beta}$. We denote ${}_{1}M_{\beta}:={}_{\operatorname{id}\nolimits}M_{\beta}$ and ${}_{\alpha}M_{1}:={}_{\alpha}M_{\operatorname{id}\nolimits}$. Recall that we can identify an $A$-bimodule $M$ with the left (right) $A^{\textrm{e}}$-module $M$ of which the structure is defined by $(a\otimes b^{\circ})m:=amb\ (m(a\otimes b^{\circ}):=bma)$ for $a\otimes b^{\circ}\in A^{\textrm{e}}$ and $m\in M$. Using this identification, we have ${}_{\alpha\otimes\beta}M={}_{\alpha}M_{\beta}=M_{\beta\otimes\alpha}$, where we set $\alpha\otimes\beta:=\alpha\otimes\beta^{\circ}$ and $\beta\otimes\alpha=:\beta\otimes\alpha^{\circ}$. For any morphism $f:M\rightarrow N$ of $A$-bimodules and $\alpha$, $\beta\in\operatorname{Aut}\nolimits(A)$, there exists an isomorphism

which is natural in both $M$ and $N$.

In this subsection, we recall the definitions of self-injective algebras and of Frobenius algebras. Recall that a finite dimensional algebra $A$ is a self-injective algebra if $A$ is injective as a left or as a right $A$-module, or equivalently, $A$ is injective as a left and as a right $A$-module. Moreover, recall that a finite dimensional $k$-algebra $A$ is a Frobenius algebra if there exists a non-degenerate bilinear form $\langle-,-\rangle:A\otimes A\rightarrow k$ satisfying $\langle ab,c\rangle=\langle a,bc\rangle$ for $a,b$ and $c\in A$. The bilinear form gives rise to a left and a right $A$-module isomorphism

where the left and the right $A$-module structure of $D(A)$ is given by

for $a,x\in A$ and $f\in D(A)$. In particular, a Frobenius algebra is a self-injective algebra. Let $\{u_{1},\ldots,u_{r}\}$ be a $k$-basis of $A$. Then we have another $k$-basis $\{v_{1},\ldots,v_{r}\}$ such that $\langle v_{i},u_{j}\rangle=\delta_{ij}$ for all $1\leq i,j\leq r$, where $\delta_{ij}$ denotes the Kronecker delta. We call $\{v_{i}\}_{i}$ the dual basis of $\{u_{i}\}_{i}$. It is known that there exists an algebra automorphism $\nu$ of $A$ such that $\langle a,b\rangle=\langle b,\nu(a)\rangle$ for $a,b\in A$. The automorphism $\nu$ is unique, up to inner automorphism, and we call it the Nakayama automorphism of $A$. The Nakayama automorphism $\nu$ of $A$ makes the left $A$-module isomorphism (2.1) into an $A$-bimodule isomorphism

Moreover, there exists another $A$-bimodule isomorphism

where the $A$-bimodule structure of $A^{\vee}$ is given by $(afb)(x):=f(x)(b\otimes a^{\circ})$ for $f\in A^{\vee}$ and $a,b\in A$.

Our aim in this subsection is to recall the definition of complete resolutions and to characterize minimal complete resolutions in terms of minimal projective resolutions and minimal injective resolutions. Let us start with the definition of complete resolutions.

We define the G-dimension $\mathrm{G}$-$\dim_{A}M$ of a finitely generated $A$-module $M$ by

Since we are interested in self-injective algebras including Frobenius algebras, we mainly deal with self-injective algebras. For more general cases, we refer to [AM02, BJ13].

Let $A$ be a self-injective algebra. Since $A$ is injective as a left and as a right $A$-module, any finitely generated $A$-module $M$ is totally reflexive and hence of G-dimension $0$. It follows from [AM02, Theorem 3.1] that $M$ has a complete resolution $\operatorname{\mathbf{T}}\nolimits\xrightarrow{\vartheta}\operatorname{\mathbf{P}}\nolimits\rightarrow M$ with $\vartheta_{i}$ isomorphic for $i\geq 0$. Thus, any complete resolution $\operatorname{\mathbf{T}}\nolimits$ of $M$ consists of some projective resolution of $M$ in non-negative degrees, and we have $M=\operatorname{Coker}\nolimits d^{\operatorname{\mathbf{T}}\nolimits}_{1}$. Thus we simply write $\operatorname{\mathbf{T}}\nolimits\rightarrow M$ for $\operatorname{\mathbf{T}}\nolimits\rightarrow\operatorname{\mathbf{P}}\nolimits\rightarrow M$.

For a complete resolution $\operatorname{\mathbf{T}}\nolimits$ of $M$ and $i\in\operatorname{\mathbb{Z}}\nolimits$, let $d_{i}^{\operatorname{\mathbf{T}}\nolimits}=\iota_{i}\pi_{i}$ be the canonical factorization through $\operatorname{Im}\nolimits d_{i}^{\operatorname{\mathbf{T}}\nolimits}$, i.e., $\pi_{i}$ is the canonical epimorphism $T_{i}\rightarrow\operatorname{Im}\nolimits d_{i}^{\operatorname{\mathbf{T}}\nolimits}$ and $\iota_{i}$ is the canonical inclusion $\operatorname{Im}\nolimits d_{i}^{\operatorname{\mathbf{T}}\nolimits}\hookrightarrow T_{i-1}$. In particular, we denote by $\varepsilon$ the epimorphism $\pi_{0}:T_{0}\rightarrow\operatorname{Im}\nolimits d_{0}^{\operatorname{\mathbf{T}}\nolimits}=M$ and by $\eta$ the canonical inclusion $\iota_{0}:M\hookrightarrow T_{-1}$. The morphism $\varepsilon:T_{0}\rightarrow M$ is called the augmentation of $\operatorname{\mathbf{T}}\nolimits$. Note that the augmentation of any complete resolution of $M$ induce a chain map $\operatorname{\mathbf{T}}\nolimits\rightarrow M$, but it is not a quasi-isomorphism. A chain map $f:\operatorname{\mathbf{T}}\nolimits\rightarrow\operatorname{\mathbf{T}}\nolimits^{\prime}$ between two complete resolutions $\operatorname{\mathbf{T}}\nolimits\xrightarrow{\varepsilon}M$ and $\operatorname{\mathbf{T}}\nolimits^{\prime}\xrightarrow{\varepsilon^{\prime}}M$ is called augmentation-preserving if $\varepsilon^{\prime}f=\varepsilon$. It follows from [AM02, Lemma 5.3] that any augmentation-preserving chain map $f:\operatorname{\mathbf{T}}\nolimits\rightarrow\operatorname{\mathbf{T}}\nolimits^{\prime}$ between two complete resolutions $\operatorname{\mathbf{T}}\nolimits$ and $\operatorname{\mathbf{T}}\nolimits^{\prime}$ of $M$ is a homotopy equivalence.