K\"ulshammer ideals of graded categories and Hochschild cohomology

TL;DR

This paper extends K"ulshammer ideals to graded categories, exploring their properties in the graded center and Hochschild cohomology, and introduces new derived invariants, especially in the context of $d$-Calabi-Yau categories.

Contribution

It generalizes K"ulshammer ideals to graded categories and studies their properties, providing new tools for invariants in derived and Hochschild cohomology.

Findings

K"ulshammer ideals are extended to graded categories.

Properties of these ideals are analyzed in the graded center and Hochschild cohomology.

Special properties are identified for $d$-Calabi-Yau categories.

Abstract

We generalize the notion of K\"ulshammer ideals to the setting of a graded category. This allows us to define and study some properties of K\"ulshammer type ideals in the graded center of a triangulated category and in the Hochschild cohomology of an algebra, providing new derived invariants. Further properties of K\"ulshammer ideals are studied in the case where the category is $d$-Calabi-Yau.

Equations

$$K_r^{cl}=\{a\in \Lambda |(a,b)=0\text{mbox}forb\in \Lambda \text{mbox}suchthat{b}^{p^r}\in [\Lambda ,\Lambda ]\}$$

$$K_{r,d}(\mathrm{A}/\mathrm{\Sigma })=\{a\in (\mathrm{A}/\mathrm{\Sigma }{)}^{\mathrm{A}/\mathrm{\Sigma }}|(a,b)=0\text{mbox}forb\in {\oplus }_{x\in \mathrm{A}}(\mathrm{A}/\mathrm{\Sigma }{)}_x^x\text{mbox}suchthat{b}^{p^r}\in [\mathrm{A}/\mathrm{\Sigma },\mathrm{A}/\mathrm{\Sigma }]\}.$$

$${\mathrm{HK}}_{r,0}^0(\Lambda )=\{a\in Z(\Lambda )|ab\in [\Lambda ,\Lambda ]\text{mbox}forb\text{mbox}suchthat{b}^{p^r}\in [\Lambda ,\Lambda ]\}.$$

$$(\mathrm{A}\otimes \mathrm{B}{)}_{(x_1,y_1)}^{(x_2,y_2)}={\mathrm{A}}_{x_1}^{x_2}\otimes {\mathrm{B}}_{y_1}^{y_2}.$$

$$(f_2\otimes g_2)(f_1\otimes g_1)=f_2{f}_1\otimes g_2{g}_1,$$

$$\mathrm{M}(y,x)\times \mathrm{M}(x,z)={\mathrm{M}}_x^y\times {\mathrm{M}}_z^x\stackrel{-\circ -}{\to }{\mathrm{M}}_z^y=\mathrm{M}(y,z)$$

$$f(v\circ u)g=(fv)\circ (ug)\text{mbox}andu\circ (fv)=(uf)\circ v,$$

$$\mathrm{M}(y,x)\times \mathrm{M}(x,z)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$m_{n,x}=(-1{)}^n{\mathrm{\Sigma }}^{-1}(m_{n,y})=(-1{)}^n{\mathrm{\Sigma }}^{-1}(m_{n,\mathrm{\Sigma }x})$$

$$fg-{\mathrm{\Sigma }}^n(g)f=(\mathrm{A}/\mathrm{\Sigma })({\mathrm{Id}}_y\otimes g)(f)-(\mathrm{A}/\mathrm{\Sigma })(g\otimes {\mathrm{Id}}_x)(f)$$

$${\mathrm{\Sigma }}^i(g)f-(-1{)}^{ij}{\mathrm{\Sigma }}^n(g){\mathrm{\Sigma }}^j(f)={\mathrm{\Sigma }}^i(g)f-(-1{)}^{j(n+1)}{\mathrm{\Sigma }}^j({\mathrm{\Sigma }}^i(g)f)\in {\mathrm{V}}_n,$$

Full text

1 Introduction

Let $\Lambda$ be a symmetric algebra over a field of positive characteristic $p$ with a symmetrizing form $(-,-)$. The sequence of Külshammer ideals

[TABLE]

in the center of $\Lambda$ is a fine invariant of the derived category of an algebra [Zim1, KLZ]. These ideals were applied to distinguish various algebras up to derived equivalence [Hol1, Hol2]. With the use of trivial extensions the definition of Külshammer ideals was extended to arbitrary algebras [BZ]. Also various attempts to generalize Külshammer ideals to higher Hochschild (co)homology were taken (see [Zim2, Zim3]).

In this paper we propose to consider the same type of ideals in the center of a graded category ${\rm A}$. These ideals are defined using the module structure of the graded abelianization of the category ${\rm A}$ over its graded center. For a graded category ${\rm A}$ the center of ${\rm A}$ is the graded $\mathbf{k}$-algebra ${\rm A}^{{\rm A}}$, whose $i$-th component ${\rm A}^{{\rm A}}_{i}$ is formed by elements $(m_{i,x})_{x\in{\rm A}}\in\Pi_{x\in{\rm A}}{\rm A}(x,x)_{i}$ such that $fm_{i,x}=(-1)^{ij}m_{i,y}f$ for any $x,y\in{\rm A}$ and any $f\in{\rm A}(x,y)_{j}$. The abelianization of ${\rm A}$ is the graded $\mathbf{k}$-module ${\rm A}_{{\rm A}}=\oplus_{x\in{\rm A}}{\rm A}_{x}^{x}/[{\rm A},{\rm A}]$, where $[{\rm A},{\rm A}]$ denotes the subspace of $\oplus_{x\in{\rm A}}{\rm A}(x,x)$ formed by the elements $fu-(-1)^{ij}uf$, for all $x,y\in{\rm A}$, $u\in{\rm A}(y,x)_{i}$ and $f\in{\rm A}(y,x)_{j}$.

The ideals $K_{r,s}{\rm A}$ are defined as the annihilators of the appropriate homogeneous component of the kernel of the map $(-)^{p^{r}}$. For precise definitions see Sections 2 and LABEL:Kidealinc. It turns out that the ideals defined in this way are invariant under graded equivalences.

This construction is then applied in the particular situation of a category ${\rm A}$ with an automorphism ${\rm\Sigma}$. In this case the orbit category ${\rm A}/{\rm\Sigma}$ is graded and we can consider ideals in the graded center of ${\rm A}/{\rm\Sigma}$. If ${\rm A}$ is $d$-Calabi-Yau, we establish a duality between the Hochshild-Mitchel homology and cohomology of ${\rm A}$. This generalizes the well known duality between Hochschild homology and cohomology for symmetric algebras. Using the pairing provided by this duality we recover the usual definition of the ideals $K_{r,d}({\rm A}/{\rm\Sigma})$ in this general context. Thus,

[TABLE]

If ${\rm A}$ is the homotopy category of complexes of finitely generated projective modules ${\rm K^{b}_{p}}\Lambda$ over some algebra $\Lambda$ and $[1]$ is the shift functor, then $({\rm K^{b}_{p}}\Lambda/[1])^{{\rm K^{b}_{p}}\Lambda/[1]}$ is the graded center ${\rm Z}^{*}({\rm K^{b}_{p}}\Lambda)$ of ${\rm K^{b}_{p}}\Lambda$, i.e. the graded ring of natural transformations from the identity functor to $[n]$ which commute modulo the sign $(-1)^{n}$ with $[1]$. Graded centers of triangulated categories are studied and used by various authors and attract some interest (see [BF, Li, BIK, KY]). Thus, the ideals defined for ${\rm K^{b}_{p}}\Lambda/[1]$ belong to the graded center of ${\rm K^{b}_{p}}\Lambda$.

The characteristic map from the Hochshild cohomology of $\Lambda$ to the graded center of ${\rm K^{b}_{p}}\Lambda$ allows us to define Külshammer ideals ${\rm HK}_{r,s}(\Lambda)$ in the higher Hochshild cohomology as the inverse image of the ideals $K_{r,s}({\rm K^{b}_{p}}\Lambda/[1])$. If $\Lambda$ is a symmetric algebra, then ${\rm HK}^{0}_{r,0}(\Lambda)$ coincide with the classical Külshammer ideals $K^{cl}_{r}$. So the ideals ${\rm HK}_{r,s}(\Lambda)$ can be considered as a generalization of Külshammer ideals to higher Hochshild cohomology. Both ideals $K_{r,s}({\rm K^{b}_{p}}\Lambda/[1])$ and ${\rm HK}_{r,s}(\Lambda)$ are invariant under derived equivalence.

For arbitrary algebras we provide an alternative description of ${\rm HK}^{0}_{r,0}(\Lambda)$.

[TABLE]

This provides an alternative generalization of Külshammer ideals for non-symmetric algebras.

We finish the paper computing all the defined ideals for the algebra $k[x]/x^{2}$.

2 Graded center and abelianization

All categories and functors are assumed to be $\mathbf{k}$-linear for some fixed field $\mathbf{k}$. Moreover, all categories are assumed to be small. We write simply $\otimes$ and ${\rm Hom}$ instead of $\otimes_{\mathbf{k}}$ and ${\rm Hom}_{\mathbf{k}}$. In this section we recall some definitions and introduce some notation. From here on we will write ${\rm A}_{x}^{y}$ instead of ${\rm Hom}_{{\rm A}}(x,y)$ for ${\rm Hom}$-sets in a category ${\rm A}$.

Definition 2.1**.**

A category ${\rm A}$ is called a graded category if, for any $x$, $y$ in ${\rm A}$, there is a fixed decomposition of graded spaces ${\rm A}_{x}^{y}=\oplus_{i\in\mathbb{Z}}({\rm A}_{x}^{y})_{i}$ such that $fg\in({\rm A}_{x}^{z})_{i+j}$ for any $f\in({\rm A}_{y}^{z})_{j}$ and $g\in({\rm A}_{x}^{y})_{i}$. We write $|f|$ for the degree of $f\in{\rm A}_{x}^{y}$, i.e. $|f|=i$ if and only if $f\in({\rm A}_{x}^{y})_{i}$. **

Definition 2.2**.**

The tensor product ${\rm A}\otimes{\rm B}$ of categories ${\rm A}$ and ${\rm B}$ is a $\mathbf{k}$-linear category defined in the following way. Its objects are pairs $(x,y)$ where $x\in{\rm A}$, $y\in{\rm B}$. Its morphism spaces are

[TABLE]

The composition in ${\rm A}\otimes{\rm B}$ is given by the formula

[TABLE]

where $f_{1}\in{\rm A}_{x_{1}}^{x_{2}}$, $f_{2}\in{\rm A}_{x_{2}}^{x_{3}}$, $g_{1}\in{\rm B}_{y_{1}}^{y_{2}}$, $g_{2}\in{\rm B}_{y_{2}}^{y_{3}}$ and $x_{1},x_{2},x_{3}\in{\rm A}$, $y_{1},y_{2},y_{3}\in{\rm B}$.**

Definition 2.3**.**

$\mathbf{k}$-linear contravariant functors from ${\rm A}$ to ${\rm Mod}\mbox{-}\mathbf{k}$ are called ${\rm A}$-modules. We denote by ${\rm Mod}\mbox{-}{\rm A}$ the category of ${\rm A}$-modules. An ${\rm A}$-bimodule is by definition an ${\rm A}^{\rm op}\otimes{\rm A}$-module. We denote by ${\rm Bimod}\mbox{-}{\rm A}$ the category of ${\rm A}$-bimodules. **

If it does not cause any confusion, we will write $fug$ instead of ${\rm M}(f\otimes g)u$ for an ${\rm A}$-bimodule ${\rm M}$ and $g\in{\rm A}_{z}^{w}$, $f\in{\rm A}_{x}^{y}$, $u\in{\rm M}(x,w)$, where $w,x,y,z\in{\rm A}$.

If ${\rm A}$ is graded, then the ${\rm A}$-bimodule ${\rm M}$ is called graded if there are some fixed decompositions ${\rm M}(x,y)=\oplus_{i\in\mathbb{Z}}{\rm M}(x,y)_{i}$ such that $fug\in{\rm M}(y,z)_{i+j+l}$ for all $g\in({\rm A}_{z}^{w})_{j}$, $f\in({\rm A}_{x}^{y})_{i}$ and $u\in{\rm M}(x,w)_{l}$. A homogeneous morphism of degree $m$ from a graded bimodule ${\rm M}$ to a graded bimodule ${\rm N}$ is a collection of maps $\phi_{x,y,k}:{\rm M}(x,y)_{k}\rightarrow{\rm N}(x,y)_{k+m}$ such that ${\rm N}(f\otimes g)\phi_{x,w,k}=(-1)^{mi}\phi_{y,z,k+i+j}{\rm M}(f\otimes g)$ for $g\in({\rm A}_{z}^{w})_{j}$, $f\in({\rm A}_{x}^{y})_{i}$. A morphism between graded bimodules is by definition a finite sum of homogeneous morphisms. Any nongraded category ${\rm A}$ can be considered as a graded category with all morphisms of degree 0. In this case a graded bimodule is simply a module ${\rm M}$ with ${\rm A}$-bimodule decomposition ${\rm M}=\oplus_{i\in\mathbb{Z}}{\rm M}_{i}$. We define the graded bimodule ${\rm M}[n]$ as a graded ${\rm A}$-bimodule such that ${\rm M}[n](x,y)_{i}={\rm M}(x,y)_{i+n}$ and ${\rm M}[n](f\otimes g)u=(-1)^{in}{\rm M}(f\otimes g)u$ for all $g\in({\rm A}_{z}^{w})_{j}$, $f\in({\rm A}_{x}^{y})_{i}$ and $u\in{\rm M}(x,w)_{l}$. If $\phi_{x,y,j}$ is a homogeneous morphism form ${\rm M}$ to ${\rm N}$, then $\phi_{x,y,j}[n]:=\phi_{x,y,j+n}$.

One can consider ${\rm A}$ as an ${\rm A}$-bimodule defined by the equality ${\rm A}(x,y)={\rm A}_{y}^{x}$ on objects and in the obvious way on morphisms.

Definition 2.4**.**

An ${\rm A}$-linear category is an ${\rm A}$-bimodule ${\rm M}$ together with a structure of a $\mathbf{k}$-linear category (which will be also denoted by ${\rm M}$) compatible with the bimodule structure. Namely, the class of objects of ${\rm M}$ is the same as the class of objects of ${\rm A}$, and the morphism spaces of ${\rm M}$ are ${\rm M}_{y}^{x}={\rm M}(x,y)$. Thus, there are bilinear maps

[TABLE]

that satisfy all the conditions of a categorical composition. The compatibility conditions are

[TABLE]

[TABLE]

where $f\in{\rm A}_{y}^{w}$ and $g\in{\rm A}_{\tau}^{z}$ are morphisms in ${\rm A}$ and $u$ and $v$ are morphisms in ${\rm M}$. **

We say that an ${\rm A}$-linear category ${\rm M}$ is graded if ${\rm M}$ is a graded category and a graded bimodule with respect to the same family of decompositions ${\rm M}^{x}_{y}=\oplus_{i\in\mathbb{Z}}({\rm M}^{x}_{y})_{i}$.

Definition 2.5**.**

Let ${\rm A}$ be a graded category and let ${\rm M}$ be a graded ${\rm A}$-bimodule. The ${\rm A}$-center of ${\rm M}$ is the graded $\mathbf{k}$-module ${\rm M}^{{\rm A}}$, whose $i$-th component ${\rm M}^{{\rm A}}_{i}$ is formed by such elements $(m_{i,x})_{x\in{\rm A}}\in\Pi_{x\in{\rm A}}{\rm M}(x,x)_{i}$ that $fm_{i,x}=(-1)^{ij}m_{i,y}f$ for any $x,y\in{\rm A}$ and any $f\in({\rm A}_{x}^{y})_{j}$. If ${\rm A}$ and ${\rm M}$ are not graded, then we can consider them as a graded category and a bimodule concentrated in degree 0. So we can talk about the center of a nongraded bimodule over a nongraded category. **

Note that Definition 2.4 guarantees that if ${\rm M}$ is an ${\rm A}$-linear category, then ${\rm M}^{{\rm A}}$ has a structure of a unital associative $\mathbf{k}$-algebra, with multiplication induced by the composition in ${\rm M}$. In particular, the compatibility of the the bimodule and the categorical structure ensures that ${\rm M}^{{\rm A}}$ is closed under multiplication.

Definition 2.6**.**

Let ${\rm A}$ be a (nongraded) category and ${\rm\Sigma}:{\rm A}\rightarrow{\rm A}$ be an automorphism of ${\rm A}$. The orbit category ${\rm A}/{\rm\Sigma}$ is a graded category defined as follows.

• •

The class of objects of ${\rm A}/{\rm\Sigma}$ is equal to that of ${\rm A}$;

• •

The sets of morphisms are \big{(}({\rm A}/{\rm\Sigma})_{x}^{y}\big{)}_{n}={\rm A}_{x}^{{\rm\Sigma}^{n}y} for $x,y\in{\rm A}/{\rm\Sigma}$ and $n\in\mathbb{Z}$;

• •

The composition $\circ$ in ${\rm A}/{\rm\Sigma}$ is given by the formula $g\circ f={\rm\Sigma}^{n}(g)f$ for f\in\big{(}({\rm A}/{\rm\Sigma})_{x}^{y}\big{)}_{n} and g\in\big{(}({\rm A}/{\rm\Sigma})_{y}^{z}\big{)}_{m}.

Note that ${\rm A}/{\rm\Sigma}$ becomes a graded ${\rm A}$-linear category if we define $({\rm A}/{\rm\Sigma})(f\otimes g)u={\rm\Sigma}^{n}(f)ug$ for $f\in{\rm A}_{y_{1}}^{y_{2}}$, $g\in{\rm A}_{x_{1}}^{x_{2}}$ and u\in\big{(}({\rm A}/{\rm\Sigma})_{x_{2}}^{y_{1}}\big{)}_{n}. Given an automorphism $\alpha$ of the category ${\rm A}$ and ${\rm M}\in{\rm Bimod}\mbox{-}{\rm A}$, we define the bimodule ${}^{\alpha}{\rm M}$ as the composition of functors ${\rm M}\circ(\alpha\otimes{\rm Id}_{{\rm A}})\in{\rm Bimod}\mbox{-}{\rm A}$. It is easy to see that this defines an action of $(\operatorname{Aut}{\rm A})^{\rm op}$ on ${\rm Bimod}\mbox{-}{\rm A}$. Note that {\rm A}/{\rm\Sigma}\cong\oplus_{n\in\mathbb{Z}}\big{(}{}^{{\rm\Sigma}^{n}}{\rm A}\big{)}[-n] as a graded ${\rm A}$-bimodule.

If $\alpha$ is an automorphism of ${\rm A}$, then we say that $\alpha$ acts on the graded bimodule ${\rm M}$ if there is a homogeneous isomorphism ${\rm M}\xrightarrow{\alpha_{{\rm M}}}{\rm M}\circ(\alpha\otimes\alpha)$ of degree [math]. Such action induces an automorphism $\alpha_{{\rm M}^{{\rm A}}}$ of the graded center ${\rm M}^{{\rm A}}$. If ${\rm M}$ is an ${\rm A}$-linear category and $\alpha$ acts on it by a category automorphism, then $\alpha_{{\rm M}^{{\rm A}}}$ is an automorphism of the graded algebra ${\rm M}^{{\rm A}}$. Note that if $\alpha$ acts on ${\rm M}$ by $\alpha_{{\rm M}}$, then we can define $\alpha_{{\rm M}[n]}$ as $\alpha_{{\rm M}[n]}=(-1)^{n}\alpha_{{\rm M}}[n]$. The automorphism ${\rm\Sigma}$ of ${\rm A}$ acts on ${}^{{\rm\Sigma}^{n}}{\rm A}$ by the rule ${\rm\Sigma}_{(^{{\rm\Sigma}^{n}}{\rm A})}(u)={\rm\Sigma}(u)$ for $u\in{\rm A}_{x}^{{\rm\Sigma}^{n}y}$. We fix this action of ${\rm\Sigma}$ on ${}^{{\rm\Sigma}^{n}}{\rm A}$ and write simply ${\rm\Sigma}$ instead of ${\rm\Sigma}_{(^{{\rm\Sigma}^{n}}{\rm A})}$. Moreover, for any integer $m$, we denote by ${\rm\Sigma}$ the natural transformation ${\rm\Sigma}_{({}^{{\rm\Sigma}^{n}}{\rm A})[m]}$ and the automorphism {\rm\Sigma}_{\big{(}({}^{{\rm\Sigma}^{n}}{\rm A})[m]\big{)}^{{\rm A}}}. Thus, ${\rm\Sigma}$ acts on the bimodule ${\rm A}/{\rm\Sigma}$ and this action determines an automorphism of the graded ${\rm A}$-linear category ${\rm A}/{\rm\Sigma}$. For any space ${\rm V}$ with an action of some automorphism $\alpha$ we can consider the subspace of invariants ${\rm V}^{\alpha}$, i.e $\{m\in{\rm V}|\alpha(m)=m\}$.

Definition 2.7**.**

Let ${\rm A}$ be a (nongraded) category and let ${\rm\Sigma}:{\rm A}\rightarrow{\rm A}$ be some fixed automorphism of ${\rm A}$. We define the graded rings ${\rm Nat}^{*}({\rm A})={\rm Nat}^{*}({\rm A},{\rm\Sigma})$ and ${\rm Z}^{*}({\rm A})={\rm Z}^{*}({\rm A},{\rm\Sigma})$ as follows. Let ${\rm Nat}^{n}({\rm A})$ ($n\in\mathbb{Z}$) be the abelian group formed by all natural transformations $\eta:{\rm Id}_{{\rm A}}\rightarrow{\rm\Sigma}^{n}$ and ${\rm Z}^{n}({\rm A})$ be its subgroup formed by $\eta$ that satisfy the equality $\eta{\rm\Sigma}=(-1)^{n}{\rm\Sigma}\eta$. Given natural transformations $\eta:{\rm Id}_{{\rm A}}\rightarrow{\rm\Sigma}^{n}$ and $\theta:{\rm Id}_{{\rm A}}\rightarrow{\rm\Sigma}^{m}$ ($n,m\in\mathbb{Z}$ ), we define the product of $\eta$ and $\theta$ by the formula $\eta\theta={\rm\Sigma}^{m}(\eta)\circ\theta:{\rm Id}_{{\rm A}}\rightarrow{\rm\Sigma}^{n+m}$. We call ${\rm Z}^{*}({\rm A})$ the graded center of ${\rm A}$. Note that $\eta\theta=(-1)^{mn}\theta\eta$ if $\eta\in{\rm Z}^{*}({\rm A})$. **

Remark 2.8**.**

The definition of a graded center has sense if ${\rm\Sigma}$ is an autoequivalence, but further we need it to be an automorphism. On the other hand, we can replace any autoequivalence of a category by an automorphism (the category is changed during this process) by the results of [Asashiba].

Lemma 2.9**.**

There is an isomorphism of graded algebras ${\rm Nat}^{*}({\rm A})\cong({\rm A}/{\rm\Sigma})^{{\rm A}}$ that induces isomorphisms {\rm Z}^{*}({\rm A})\cong\Big{(}({\rm A}/{\rm\Sigma})^{{\rm A}}\Big{)}^{{\rm\Sigma}}\cong({\rm A}/{\rm\Sigma})^{{\rm A}/{\rm\Sigma}}.

• Proof.

By definition, ${\rm A}$-center of ${\rm A}/{\rm\Sigma}$ has as the $n$-th component \big{(}({\rm A}/{\rm\Sigma})^{{\rm A}}\big{)}_{n} elements $(m_{n,x})_{x\in{\rm A}}\in\Pi_{x\in{\rm A}}({\rm A}/{\rm\Sigma})(x,x)_{n}$ such that ${\rm\Sigma}^{n}(f)m_{n,x}=m_{n,y}f$ for any $x,y\in{\rm A}$ and any $f\in{\rm A}_{x}^{y}$. Thus, a family $(m_{n,x})_{x\in{\rm A}}$ belongs to \big{(}({\rm A}/{\rm\Sigma})^{{\rm A}}\big{)}_{n} if and only if it gives a natural transformation ${\rm Id}_{{\rm A}}\rightarrow{\rm\Sigma}^{n}$. The multiplication on ${\rm Nat}^{*}({\rm A})$ is $\eta\theta={\rm\Sigma}^{m}(\eta)\circ\theta$, the multiplication on $({\rm A}/{\rm\Sigma})^{{\rm A}}$ is induced by the composition in ${\rm A}/{\rm\Sigma}$, i.e. $gf={\rm\Sigma}^{n}(g)\circ f$, hence these graded algebras are isomorphic.

As for the isomorphism ${\rm Z}^{*}({\rm A})\cong({\rm A}/{\rm\Sigma})^{{\rm A}/{\rm\Sigma}}$, it is clear that an element of \big{(}({\rm A}/{\rm\Sigma})^{{\rm A}/{\rm\Sigma}}\big{)}_{n} is a natural transformation, as in the previous paragraph, i.e. $({\rm A}/{\rm\Sigma})^{{\rm A}/{\rm\Sigma}}$ is a subalgebra of $({\rm A}/{\rm\Sigma})^{{\rm A}}$. Taking $y={\rm\Sigma}x$ and f={\rm Id}_{x}\in{\rm A}_{x}^{{\rm\Sigma}^{-1}y}=\big{(}({\rm A}/{\rm\Sigma})_{x}^{y}\big{)}_{-1} we get

[TABLE]

and hence ${\rm\Sigma}(m_{n,x})=(-1)^{n}m_{n,{\rm\Sigma}x}$; thus, we get an element of ${\rm Z}^{*}({\rm A})$. If we take an element $\eta$ of ${\rm Z}^{n}({\rm A})$, then for $f:x\rightarrow{\rm\Sigma}^{k}y$ we get ${\rm\Sigma}^{n}(f)\eta_{x}=\eta_{{\rm\Sigma}^{k}y}f=(-1)^{nk}{\rm\Sigma}^{k}(\eta_{y})f$; thus, we get an element of \big{(}({\rm A}/{\rm\Sigma})^{{\rm A}/{\rm\Sigma}}\big{)}_{n}.

The isomorphism ${\rm Z}^{*}({\rm A})\cong(({\rm A}/{\rm\Sigma})^{{\rm A}})^{{\rm\Sigma}}$ follows from the definition of the action of ${\rm\Sigma}$ on the bimodule ${\rm A}/{\rm\Sigma}$: the family $\{m_{n,x}\}_{x\in{\rm A}}$ belongs to $(({\rm A}/{\rm\Sigma})^{{\rm A}})^{{\rm\Sigma}}$ if and only if $(-1)^{n}{\rm\Sigma}(m_{m,x})=m_{n,{\rm\Sigma}x}$ if and only if the corresponding natural transformation belongs to ${\rm Z}^{*}({\rm A})$.

$\Box$

Definition 2.10**.**

Given a graded ${\rm A}$-bimodule ${\rm M}$, $[{\rm A},{\rm M}]$ denotes the subspace of $\oplus_{x\in{\rm A}}{\rm M}_{x}^{x}$ formed by the elements $fu-(-1)^{ij}uf$, for all $x,y\in{\rm A}$, $u\in{\rm M}(y,x)_{i}$ and $f\in({\rm A}_{y}^{x})_{j}$. The ${\rm A}$-abelianization of ${\rm M}$ is the graded $\mathbf{k}$-module ${\rm M}_{{\rm A}}=\oplus_{x\in{\rm A}}{\rm M}_{x}^{x}/[{\rm A},{\rm M}]$. As in the case of the center, we can talk about the abelianization of a nongraded bimodule over a nongraded category. **

For any space ${\rm V}$ with an action of some automorphism $\alpha$ we can consider the space of co-invariants ${\rm V}_{\alpha}$, i.e. the quotient space of ${\rm V}$ modulo the subspace generated by the classes of the elements of the form $v-\alpha(v)$ for $v\in{\rm V}$. If ${\rm M}$ is an ${\rm A}$-linear category, then the composition in ${\rm M}$ induces a structure of a graded ${\rm M}^{{\rm A}}$-bimodule on ${\rm M}_{{\rm A}}$. Note that an action of an automorphism on ${\rm M}$ induces an action on ${\rm M}_{{\rm A}}$. So in this case one can define the graded space $({\rm M}_{{\rm A}})_{{\rm\Sigma}}$ as the quotient space of ${\rm M}_{{\rm A}}$ modulo the subspace generated by the classes of the elements of the form $m-{\rm\Sigma}m$ for $m\in\oplus_{x\in{\rm A}}{\rm M}_{x}^{x}$. The ${\rm M}^{{\rm A}}$-bimodule structure on ${\rm M}_{{\rm A}}$ induces the $({\rm M}^{{\rm A}})^{{\rm\Sigma}}$-bimodule structure on $({\rm M}_{{\rm A}})_{{\rm\Sigma}}$. Note also that if the action of the automorphism $\alpha$ of ${\rm A}$ on the bimodule ${\rm M}$ is given by $\alpha_{{\rm M}}$, then $\alpha$ acts on ${\rm M}$ by $a\alpha_{{\rm M}}$ for any $a\in\mathbf{k}^{*}$ as well.

Definition 2.11**.**

Let ${\rm A}$ be a (nongraded) category and let ${\rm\Sigma}:{\rm A}\rightarrow{\rm A}$ be some fixed automorphism of ${\rm A}$. We define the graded spaces ${\rm Nat}_{*}({\rm A})={\rm Nat}_{*}({\rm A},{\rm\Sigma})$ and ${\rm Ab}_{*}({\rm A})={\rm Ab}_{*}({\rm A},{\rm\Sigma})$ as follows. Let ${\rm Nat}_{n}({\rm A})$ be the quotient space of the space $\oplus_{x\in{\rm A}}{\rm A}_{x}^{{\rm\Sigma}^{n}x}$ modulo the subspace generated by the elements of the form $fg-{\rm\Sigma}^{n}(g)f$ for all $f\in{\rm A}_{x}^{{\rm\Sigma}^{n}y}$ and $g\in{\rm A}_{y}^{x}$. We define ${\rm Ab}_{n}({\rm A})$ as the quotient space of ${\rm Nat}_{n}({\rm A})$ modulo the subspace generated by the classes of the elements of the form $f+(-1)^{n}{\rm\Sigma}(f)$ for all $f\in{\rm A}_{x}^{{\rm\Sigma}^{n}x}$. We call ${\rm Ab}_{*}({\rm A})$ the graded abelianization of ${\rm A}$. **

It is easy to see that ${\rm Nat}_{*}({\rm A})$ is a ${\rm Nat}^{*}({\rm A})$-bimodule. Moreover, the corresponding ${\rm Z}^{*}({\rm A})$-bimodule structure on ${\rm Nat}_{*}({\rm A})$ induces a ${\rm Z}^{*}({\rm A})$-bimodule structure on ${\rm Ab}_{*}({\rm A})$. Note also that the isomorphisms from Lemma 2.9 induce a ${\rm Nat}^{*}({\rm A})$-bimodule structure on $({\rm A}/{\rm\Sigma})_{{\rm A}}$ and a ${\rm Z}^{*}({\rm A})$-bimodule structure on $({\rm A}/{\rm\Sigma})_{{\rm A}/{\rm\Sigma}}$ and \Big{(}({\rm A}/{\rm\Sigma})_{{\rm A}}\Big{)}_{-{\rm\Sigma}} (in fact, as it was noted above, on \Big{(}({\rm A}/{\rm\Sigma})_{{\rm A}}\Big{)}_{a{\rm\Sigma}} for any $a\in\mathbf{k}^{*}$).

Lemma 2.12**.**

There is an isomorphism of graded ${\rm Nat}^{*}({\rm A})$-bimodules ${\rm Nat}_{*}({\rm A})\cong({\rm A}/{\rm\Sigma})_{{\rm A}}$ that induces isomorphisms {\rm Ab}_{*}({\rm A})\cong\Big{(}({\rm A}/{\rm\Sigma})_{{\rm A}}\Big{)}_{-{\rm\Sigma}}\cong({\rm A}/{\rm\Sigma})_{{\rm A}/{\rm\Sigma}}.

• Proof.

By definition, ${\rm Nat}_{*}({\rm A})$ and $({\rm A}/{\rm\Sigma})_{{\rm A}}$ are both quotients of the ${\rm Nat}^{*}({\rm A})$-bimodule $\oplus_{n\in\mathbb{Z}}\oplus_{x\in{\rm A}}{\rm A}_{x}^{{\rm\Sigma}^{n}x}$ and it suffices to check that we take the quotient space modulo the same submodule.

It is easy to see that

[TABLE]

for all $f\in{\rm A}_{x}^{{\rm\Sigma}^{n}y}$ and $g\in{\rm A}_{y}^{x}$, i.e. {\rm Nat}_{n}({\rm A})=\big{(}({\rm A}/{\rm\Sigma})_{{\rm A}}\big{)}_{n}=\oplus_{x\in{\rm A}}{\rm A}_{x}^{{\rm\Sigma}^{n}x}/{\rm U}_{n}, where ${\rm U}_{n}$ is generated by $fg-{\rm\Sigma}^{n}(g)f$ for $f\in{\rm A}_{x}^{{\rm\Sigma}^{n}y}$ and $g\in{\rm A}_{y}^{x}$.

Now we have ${\rm Ab}_{n}({\rm A})=\oplus_{x\in{\rm A}}{\rm A}_{x}^{{\rm\Sigma}^{n}x}/({\rm U}_{n}+{\rm V}_{n})$, where ${\rm V}_{n}$ is generated by $f+(-1)^{n}{\rm\Sigma}(f)$ for all $f\in{\rm A}_{x}^{{\rm\Sigma}^{n}x}$. It is easy to see that \Big{(}\big{(}({\rm A}/{\rm\Sigma})_{{\rm A}}\big{)}_{-{\rm\Sigma}}\Big{)}_{n}=\oplus_{x\in{\rm A}}{\rm A}_{x}^{{\rm\Sigma}^{n}x}/({\rm U}_{n}+{\rm V}_{n}) too. Let now consider \big{(}({\rm A}/{\rm\Sigma})_{{\rm A}/{\rm\Sigma}}\big{)}_{n}=\oplus_{x\in{\rm A}}{\rm A}_{x}^{{\rm\Sigma}^{n}x}/{\rm W}_{n}, where ${\rm W}_{n}$ is generated by ${\rm\Sigma}^{j}(f)g-(-1)^{ij}{\rm\Sigma}^{i}(g)f$ for $f\in{\rm A}_{x}^{{\rm\Sigma}^{i}y}$ and $g\in{\rm A}_{y}^{{\rm\Sigma}^{j}x}$, $i+j=n$. Taking $j=0$, we get that ${\rm U}_{n}\subseteq{\rm W}_{n}$. Taking $i=n+1$, $j=-1$, $y={\rm\Sigma}^{-1}x$, and $g={\rm Id}_{x}$, we get that ${\rm V}_{n}\subseteq{\rm W}_{n}$. Thus, ${\rm U}_{n}+{\rm V}_{n}\subseteq{\rm W}_{n}$.

Since ${\rm\Sigma}^{j}(f)g-{\rm\Sigma}^{n}(g){\rm\Sigma}^{j}(f)\in{\rm U}_{n}$ and

[TABLE]

we have ${\rm W}_{n}={\rm U}_{n}+{\rm V}_{n}$.