Cohomology of modules over $H$-categories and co-$H$-categories

TL;DR

This paper develops spectral sequences for cohomology theories of modules over Hopf algebra categories, generalizing classical Hopf module concepts and providing tools for their cohomological analysis.

Contribution

It introduces a framework for cohomology of modules over $H$-categories and co-$H$-categories, including spectral sequences and generalizations of Hopf modules.

Findings

Constructed Grothendieck spectral sequences for cohomologies.

Extended the concept of Hopf modules to relative $( ext{D},H)$-Hopf modules.

Provided methods to compute cohomologies using rational $Hom$ objects.

Abstract

Let $H$ be a Hopf algebra. We consider $H$-equivariant modules over a Hopf module category $\mathcal C$ as modules over the smash extension $\mathcal C\# H$. We construct Grothendieck spectral sequences for the cohomologies as well as the $H$-locally finite cohomologies of these objects. We also introduce relative $(\mathcal D,H)$-Hopf modules over a Hopf comodule category $\mathcal D$. These generalize relative $(A,H)$-Hopf modules over an $H$-comodule algebra $A$. We construct Grothendieck spectral sequences for their cohomologies by using their rational $Hom$ objects and higher derived functors of coinvariants.

Equations

$$R^p(-{)}^H\left({\text{Ext}}_{Mod\text{-}\mathcal{C}}^q(\mathcal{M},\mathcal{N})\right)\Rightarrow \left(R^{p+q}Ho{m}_{Mod\text{-}(\mathcal{C}\#H)}(\mathcal{M},-)\right)(\mathcal{N})$$

$${\mathcal{L}}_{Mod\text{-}\mathcal{C}}:(Mod\text{-}(\mathcal{C}\#H){)}^{op}\times Mod\text{-}(\mathcal{C}\#H)⟶H\text{-}mod{\mathcal{L}}_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N}):=Ho{m}_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N}{)}^{(H)}$$

$$R^p{(-)}^{(H)}({\text{Ext}}_{Mod\text{-}\mathcal{C}}^q(\mathcal{M},\mathcal{N}))\Rightarrow \left({\mathbf{R}}^{p+q}{\mathcal{L}}_{Mod\text{-}\mathcal{C}}(\mathcal{M},-)\right)(\mathcal{N})$$

$$R^{p}(-)^{H}\big{(}\textnormal{Ext}^{q}_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})\big{)}\Rightarrow\left(R^{p+q}Hom_{mod\text{-}(\mathcal{C}\#H)}(\mathcal{M},-)\right)(\mathcal{N})$$

$$HO{M}_{\mathcal{D}\text{-}Mod}:({}_{\mathcal{D}}{\mathcal{M}}^H{)}^{op}\times {}_{\mathcal{D}}{\mathcal{M}}^H⟶Comod\text{-}H$$

$$R^p(-{)}^{coH}(R^{q}HO{M}_{\mathcal{D}\text{-}Mod}(\mathcal{M},-)(\mathcal{N}))\Rightarrow \left(R^{p+q}Ho{m}_{{}_{\mathcal{D}}{\mathcal{M}}^H}(\mathcal{M},-)\right)(\mathcal{N})$$

$$R^{p}(-)^{coH}\big{(}\textnormal{Ext}^{q}_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{N})\big{)}\Rightarrow\left({R}^{p+q}Hom_{{{}_{\mathcal{D}}}{\mathscr{M}}^{H}}(\mathcal{M},-)\right)(\mathcal{N})$$

$$h(fg)=\sum h_1(f)h_2(g)$$

$$h(f)(x)=\sum h_{1}f(S(h_2)x)\forall x\in X,\forall f\in Ho{m}_A(X,Y)$$

$${\rho }_{XY}:Ho{m}_{\mathcal{D}}(X,Y)⟶Ho{m}_{\mathcal{D}}(X,Y)\otimes H,{\rho }_{XY}(f)=\sum f_0\otimes f_1$$

$${\rho }_{XZ}(fg)=\sum {(fg)}_0\otimes {(fg)}_1=\sum f_0{g}_0\otimes f_1{g}_1={\rho }_{YZ}(f){\rho }_{XY}(g)$$

$$h^{\ast }(f):=\sum f_0{h}^{\ast }(f_1)$$

$$h^{\ast }(fg)$$

$${\rho }_{XY}:Ho{m}_{\mathcal{D}}(X,Y)⟶Ho{m}_{\mathcal{D}}(X,Y)\otimes H,{\rho }_{XY}(f):=\sum _{i=1}^n{e}_i^{\ast }(f)\otimes e_i$$

$$\begin{array}[]{ll}(id\otimes h^{*})(\rho_{XZ}(fg))=(id\otimes h^{*})\left(\sum_{i=1}^{n}e_{i}^{*}(fg)\otimes e_{i}\right)&=\sum_{i=1}^{n}e_{i}^{*}(fg)\otimes h^{*}(e_{i})\\ &=\sum_{i=1}^{n}(h^{*}(e_{i})e_{i}^{*})(fg)\otimes 1_{H}\\ &=h^{*}(fg)\otimes 1_{H}\\ &=\sum_{i=1}^{n}h_{1}^{*}(f)h_{2}^{*}(g)\otimes 1_{H}\\ &=\sum_{1\leq i,j\leq n}(h^{*}_{1}(e_{i})e_{i}^{*})(f)(h^{*}_{2}(e_{j})e_{j}^{*})(g)\otimes 1_{H}\\ &=\sum_{1\leq i,j\leq n}e_{i}^{*}(f)e_{j}^{*}(g)\otimes h^{*}(e_{i}e_{j})\\ &=(id\otimes h^{*})\left(\sum_{1\leq i,j\leq n}e_{i}^{*}(f)e_{j}^{*}(g)\otimes e_{i}e_{j}\right)\end{array}$$

$${\rho }_{XZ}(fg)=\sum _{1\le i,j\le n}{e}_i^{\ast }(f)e_j^{\ast }(g)\otimes e_i{e}_j={\rho }_{YZ}(f){\rho }_{XY}(g)$$

$$Ho{m}_{{\mathcal{D}}^{H^{\ast }}}(X,Y)$$

$$=\{f\in Ho{m}_{\mathcal{D}}(X,Y)|\sum f_0{h}^{\ast }(f_1)=h^{\ast }(1_H)f,\forall h^{\ast }\in H^{\ast }\}$$

$$=\{f\in Ho{m}_{\mathcal{D}}(X,Y)|(id\otimes h^{\ast })({\rho }_{XY}(f))=(id\otimes h^{\ast })(f\otimes 1_H),\forall h^{\ast }\in H^{\ast }\}$$

$$=\{f\in Ho{m}_{\mathcal{D}}(X,Y)|{\rho }_{XY}(f)=f\otimes 1_H\}=Ho{m}_{{\mathcal{D}}^{coH}}(X,Y).$$

$$h(\mathcal{M}(f)(m))=\sum \mathcal{M}(h_{2}f)(h_{1}m)\forall h\in H,f\in Ho{m}_{\mathcal{C}}(X,Y),m\in \mathcal{M}(Y)$$

$$(h\cdot \eta )(X)(m)=\sum h_1\eta (X)(S(h_2)m)$$

$$\begin{array}[]{ll}\sum((h_{1}\nu)(X)\circ(h_{2}\eta)(X))(m)&=\sum(h_{1}\nu)(X)\left(h_{2}\eta(X)(S(h_{3})m)\right)\\ &=\sum h_{1}\nu(X)\left(S(h_{2})h_{3}\eta(X)(S(h_{4})m)\right)\\ &=\sum h_{1}\nu(X)\left(\eta(X)(\varepsilon(h_{2})S(h_{3})m)\right)\\ &=\sum h_{1}\nu(X)\left(\eta(X)(S(h_{2})m)\right)\\ &=(h\cdot(\nu\circ\eta))(X)(m)\end{array}$$

$$\begin{array}[]{ll}\eta(X)(hm)=\sum\eta(X)(\varepsilon(h_{1})h_{2}m)=\sum\varepsilon(h_{1})\eta(X)(h_{2}m)&=\sum(h_{1}\cdot\eta)(X)(h_{2}m)\\ &=\sum h_{1}\eta(X)(S(h_{2})h_{3}m)=h\eta(X)(m)\\ \end{array}$$

$$\begin{array}[]{c}Ob(\mathcal{C}\#H):=Ob(\mathcal{C})\qquad Hom_{\mathcal{C}\#H}(X,Y):=Hom_{\mathcal{C}}(X,Y)\otimes H\\ \end{array}$$

$$(f\#h)(g\#h^{\prime })=\sum f(h_{1}g)\#(h_2{h}^{\prime })$$

$$hm:=\mathcal{M}(i{d}_X\#S(h))(m)\forall h\in H,m\in \mathcal{M}(X)$$

$$h(h^{\prime }m)$$

$$=\mathcal{M}((i{d}_X\#S(h^{\prime }))(i{d}_X\#S(h)))(m)$$

$$=\sum \mathcal{M}(S(h_2^{\prime })(i{d}_X)\#S(h_1^{\prime })S(h))(m)$$

$$=\sum \mathcal{M}(\epsilon (S(h_2^{\prime }))(i{d}_X)\#S(h_1^{\prime })S(h))(m)\text{(using Definition refdefH-cat(ii))}$$

$$=\mathcal{M}(i{d}_X\#S(h{h}^{\prime }))(m)\text{(using ε∘S=ε)}$$

$$=(h{h}^{\prime })(m)$$

$$\begin{array}[]{c}\mathcal{M}^{\prime}(X):=\mathcal{M}(X)\ \ \ \forall\ X\in Ob(\mathcal{C}\#H),\\ \mathcal{M}^{\prime}(f\#h)(m):=S^{-1}(h)\mathcal{M}(f)(m)\ \ \ \forall\ f\#h\in Hom_{\mathcal{C}\#H}(Y,X),~{}m\in\mathcal{M}(X).\end{array}$$

Full text

Department of Mathematics, Indian Institute of Science, Bangalore 560012, India.

MSC(2010) Subject Classification: 16S40, 16T05, 18E05

Keywords: $H$-categories, co-$H$-categories, $H$-equivariant modules, relative Hopf modules

1 Introduction

Let $H$ be a Hopf algebra over a field $K$. An $H$-category is a small $K$-linear category $\mathcal{C}$ such that the morphism space $Hom_{\mathcal{C}}(X,Y)$ is an $H$-module for each couple of objects $X$, $Y\in Ob(\mathcal{C})$ and the composition of morphisms in $\mathcal{C}$ is well-behaved with respect to the action of $H$. Similarly, a co-$H$-category is a small $K$-linear category $\mathcal{D}$ such that the morphism space $Hom_{\mathcal{D}}(X,Y)$ is an $H$-comodule for each couple of objects $X$, $Y\in Ob(\mathcal{D})$ and the composition of morphisms in $\mathcal{D}$ is well-behaved with respect to the coaction of $H$. In other words, an $H$-category is enriched over the monoidal category of $H$-modules and a co-$H$-category is enriched over the monoidal category of $H$-comodules. The purpose of this paper is to study cohomology in module categories over $H$-categories and co-$H$-categories.

The Hopf module categories that we use were first considered by Cibils and Solotar [CiSo], where they discovered a Morita equivalence that relates Galois coverings of a category to its smash extensions via a Hopf algebra. We view these $H$-categories and the modules over them as objects of interest in their own right. We recall here that an ordinary ring may be expressed as a preadditive category with a single object. Accordingly, an arbitrary small preadditive category may be understood as a ‘ring with several objects’ (see Mitchell [Mit1]). As such, the theories obtained by replacing rings by preadditive categories have been developed widely in the literature (see, for instance, [AB1], [EV], [LoVa], [LoVa1],[Lo],[Xu1], [Xu2]). In this respect, an $H$-category may be seen as an “$H$-module algebra with several objects”. Likewise, a co-$H$-category may be seen as an “$H$-comodule algebra with several objects.”

The various aspects of categorified Hopf actions and coactions on algebras have already been studied by several authors. In [HS07], Herscovich and Solotar obtained a Grothendieck spectral sequence for the Hochschild-Mitchell cohomology of an $H$-comodule category appearing as an $H$-Galois extension. Hopf comodule categories were also studied in [StSt], where the authors introduced cleft $H$-comodule categories and extended classical results on cleft comodule algebras. More recently, Batista, Caenepeel and Vercruysse have shown in [BCV] that several deep theorems on Hopf modules can be extended to a categorification of Hopf algebras (see also [CF]).

In this paper, we will construct a Grothendieck spectral sequence that computes the higher derived $Hom$ functors for $H$-equivariant modules over an $H$-category $\mathcal{C}$. We will also construct a spectral sequence that gives the higher derived $Hom$ functors for relative $(\mathcal{D},H)$-modules, where $\mathcal{D}$ is a co-$H$-category. We will develop these cohomology theories in a manner analogous to the “$H$-finite cohomology” obtained by Guédénon [Gue] (see also [Gue1]) and the cohomology of relative Hopf modules studied by Caenepeel and Guédénon in [CanGue] respectively.

We now describe the paper in more detail. We begin in Section 2 by recalling the notion of a left $H$-category and a right co-$H$-category. For a left $H$-category $\mathcal{C}$, we have a category of $H$-invariants which will be denoted by $\mathcal{C}^{H}$. For a right co-$H$-category $\mathcal{D}$, there is a corresponding category of $H$-coinvariants which will be denoted by $\mathcal{D}^{coH}$. If $H$ is a finite dimensional Hopf algebra and $H^{*}$ is its linear dual, then a $K$-linear category $\mathcal{D}$ is a left $H^{*}$-category if and only if it is a right co-$H$-category. In that case, $\mathcal{D}^{H^{*}}=\mathcal{D}^{coH}$.

In Sections 3 and 4, we work with a left $H$-category $\mathcal{C}$. We consider right $\mathcal{C}$-modules that are equipped with an additional left $H$-equivariant structure (see Definition 3.2). This category is denoted by $(Mod\text{-}\mathcal{C})^{H}_{H}$. If $\mathcal{M}$, $\mathcal{N}\in(Mod\text{-}\mathcal{C})^{H}_{H}$, the space $Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})$ of right $\mathcal{C}$-module morphisms carries a left $H$-module structure whose $H$-invariants are given by $Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})^{H}=Hom_{(Mod\text{-}\mathcal{C})^{H}_{H}}(\mathcal{M},\mathcal{N})$.

More precisely, let $(Mod\text{-}\mathcal{C})_{H}$ denote the category with the same objects as $(Mod\text{-}\mathcal{C})^{H}_{H}$ but whose morphisms are ordinary $\mathcal{C}$-modules morphisms. Then, we show that $(Mod\text{-}\mathcal{C})_{H}$ is a left $H$-category and $(Mod\text{-}\mathcal{C})^{H}_{H}$ may be recovered as the category of $H$-invariants of $(Mod\text{-}\mathcal{C})_{H}$. Further, we obtain that $(Mod\text{-}\mathcal{C})^{H}_{H}$ is identical to the category $Mod\text{-}(\mathcal{C}\#H)$ of right modules over the smash product category $\mathcal{C}\#H$. In particular, this shows that $(Mod\text{-}\mathcal{C})^{H}_{H}$ is a Grothendieck category. We then construct a Grothendieck spectral sequence (see Theorem 3.15)

[TABLE]

for the higher derived $Hom$ in $Mod\text{-}(\mathcal{C}\#H)$ in terms of the derived $Hom$ in $Mod\text{-}\mathcal{C}$ and the derived functor of $H$-invariants.

We proceed in Section 4 to develop the “$H$-finite cohomology” of $(\mathcal{C}\#H)$-modules in a manner analogous to Guédénon [Gue]. If $M$ is an $H$-module, we denote by $M^{(H)}$ the collection of all elements $m\in M$ such that $Hm$ is a finite dimensional vector space. In particular, $M$ is said to be $H$-locally finite if $M^{(H)}=M$ and we let $H$-$mod$ denote the category of $H$-locally finite modules. This leads to a functor

[TABLE]

We then construct a Grothendieck spectral sequence (see Theorem 4.2)

[TABLE]

The left $H$-category $\mathcal{C}$ is said to be locally finite if every morphism space $Hom_{\mathcal{C}}(X,Y)$ is locally finite as an $H$-module. We denote by $mod\text{-}(\mathcal{C}\#H)$ the full subcategory of $Mod\text{-}(\mathcal{C}\#H)$ consisting of those left $H$-equivariant right $\mathcal{C}$-modules $\mathcal{M}$ such that $\mathcal{M}(X)$ is $H$-locally finite for each $X\in Ob(\mathcal{C})$. When $\mathcal{C}$ is left $H$-locally finite and right noetherian, we construct a spectral sequence (see Theorem 4.19)

[TABLE]

In Section 5, we work with a right co-$H$-category $\mathcal{D}$ and introduce the category ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ of relative $(\mathcal{D},H)$-Hopf modules (see Definition 5.1). A relative $(\mathcal{D},H)$-module consists of an $H$-coaction on a pair $(\mathcal{D},\mathcal{M})$, where $\mathcal{M}$ is a left $\mathcal{D}$-module. In particular, $\mathcal{M}(X)$ is equipped with the structure of a right $H$-comodule for each $X\in Ob(\mathcal{D})$. We show that ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ is a Grothendieck category.

Let $Comod\text{-}H$ be the category of $H$-comodules. Thereafter, we construct a functor (see (5.3))

[TABLE]

by using the right adjoint of the functor $\mathcal{N}\otimes(-):Comod\text{-}H\longrightarrow{{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ for each fixed $\mathcal{N}\in{{}_{\mathcal{D}}}{\mathscr{M}}^{H}$. In the case of an $H$-comodule algebra as considered by Caenepeel and Guédénon, the $HOM$ functor gives the collection of “rational morphisms” between relative Hopf modules (see[CanGue, $\S$ 2]). Although the category ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ is not necessarily enriched over $Comod\text{-}H$, we see that $HOM_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{N})$ behaves like a $Hom$ object. The morphisms in $Hom_{{{}_{\mathcal{D}}}{\mathscr{M}}^{H}}(\mathcal{M},\mathcal{N})$ may be recovered as the $H$-coinvariants $HOM_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{N})^{coH}=Hom_{{{}_{\mathcal{D}}}{\mathscr{M}}^{H}}(\mathcal{M},\mathcal{N})$. We then construct a Grothendieck spectral sequence (see Theorem 5.9)

[TABLE]

For $\mathcal{M}$, $\mathcal{N}\in{{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ with $\mathcal{M}$ finitely generated as a $\mathcal{D}$-module, we show that $Hom_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{N})$ is an $H$-comodule and that $HOM_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{N})=Hom_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{N})$. When $\mathcal{D}$ is also left noetherian, we construct a Grothendieck spectral sequence (see Theorem 5.17)

[TABLE]

for the higher derived $Hom$ in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$.

Notations: Throughout the paper, $K$ is a field, $H$ is a Hopf algebra with comultiplication $\Delta$, counit $\varepsilon$ and bijective antipode $S$. We shall use Sweedler’s notation for the coproduct $\Delta(h)=\sum h_{1}\otimes h_{2}$ and for a coaction $\rho:M\longrightarrow M\otimes H$, $\rho(m)=\sum m_{0}\otimes m_{1}$. We denote by $H^{*}$ the linear dual of $H$. The category of left $H$-modules will be denoted by $H$-$Mod$ and the category of right $H$-comodules will be denoted by $Comod$-$H$. For $M\in H$-$Mod$, we set $M^{H}:=\{m\in M~{}|~{}hm=\varepsilon(h)m~{}\forall h\in H\}$. For $M\in Comod$-$H$, we set $M^{coH}:=\{m\in M~{}|~{}\rho(m)=m\otimes 1_{H}\}$.

2 $H$-categories and co-$H$-categories

Let $H$ be a Hopf algebra over a field $K$. Then, it is well known (see, for instance, [Psch1, $\S$ 2.2]) that the category of $H$-modules as well as the category of $H$-comodules is monoidal. A $K$-linear category is said to be an $H$-module category (resp. an $H$-comodule category) if it is enriched over the monoidal category of $H$-modules (resp. $H$-comodules). For more on enriched categories, the reader may see, for example, [Borc, Chapter 6] or [GMK].

Definition 2.1**.**

(see Cibils and Solotar [CiSo, Definition 2.1]) Let $K$ be a field. A $K$-linear category $\mathcal{C}$ is said to be a left $H$-module category if it is enriched over the monoidal category of left $H$-modules. In other words, it satisfies the following conditions:

• (i)

$Hom_{\mathcal{C}}(X,Y)$* is a left $H$-module for all $X,Y\in Ob(\mathcal{C})$.*

• (ii)

$h(\text{id}_{X})=\varepsilon(h)\cdot\text{id}_{X}$* for every $X\in Ob(\mathcal{C})$ and every $h\in H$.*

• (iii)

The composition of morphisms in $\mathcal{C}$ is $H$-equivariant, i.e., for any $h\in H$ and any pair of composable morphisms $g:X\longrightarrow Y$, $f:Y\longrightarrow Z$, we have

[TABLE]

By a left $H$-category, we will always mean a small left $H$-module category. A right $H$-category may be defined similarly.

Definition 2.2**.**

Let $\mathcal{C}$ be a left $H$-module category. A morphism $f\in Hom_{\mathcal{C}}(X,Y)$ is said to be $H$-invariant if $h(f)=\varepsilon(h)\cdot f$ for all $h\in H$. The subcategory whose objects are the same as those of $\mathcal{C}$ and whose morphisms are the $H$-invariant morphisms in $\mathcal{C}$ is denoted by $\mathcal{C}^{H}$.

Let $A$ be a left $H$-module algebra. A right $A$-module $M$ is said to be left $H$-equivariant if

• (i)

$M$ is a left $H$-module and

• (ii)

the action of $A$ on $M$ is a morphism of $H$-modules, i.e., $h(ma)=\sum h_{1}(m)h_{2}(a)$, for all $h\in H,~{}a\in A$ and $m\in M.$

Example 2.3**.**

(see [kk]) Let $A$ be a left $H$-module algebra.

• (i)

Then, the category ${}_{H}{M}{{}^{A}}$ of (isomorphism classes of) all left $H$-equivariant finitely generated right $A$-modules, with right $A$-module morphisms between them, is an $H$-category. In fact, one can check that for $X,Y\in Ob(_{H}{M}{{}^{A}})$, the morphism space $Hom_{A}(X,Y)$ is a left $H$-module via

[TABLE]

• (ii)

The finitely generated free right $A$-modules are automatically left $H$-equivariant. The category of (isomorphism classes of) finitely generated free right $A$-modules is an $H$-category.

We may also define the notion of a co-$H$-category, which replaces an $H$-comodule algebra (see [StSt]). This notion also appears implicitly in [CiSo].

Definition 2.4**.**

By a right co-$H$-category, we will mean a small $K$-linear category $\mathcal{D}$ that is enriched over the monoidal category of right $H$-comodules. In other words, we have:

1. (i)

$Hom_{\mathcal{D}}(X,Y)$* is a right $H$-comodule for all $X,Y\in Ob(\mathcal{D}),$ with structure map*

[TABLE] 2. (ii)

$\rho_{XX}(\text{id}_{X})=\text{id}_{X}\otimes 1_{H}$, for any $X\in Ob(\mathcal{D})$ and any $h\in H$. 3. (iii)

The composition of morphisms in $\mathcal{D}$ is $H$-coequivariant, i.e., for any pair of composable morphisms $g:X\longrightarrow Y$, $f:Y\longrightarrow Z$, we have

[TABLE]

A left co-$H$-category may be defined similarly.

A morphism $f\in Hom_{\mathcal{D}}(X,Y)$ in a right co-$H$-category is said to be $H$-coinvariant if it satisfies $\rho_{XY}(f)=f\otimes 1_{H}$. The subcategory whose objects are the same as those of $\mathcal{D}$ and whose morphisms are $H$-coinvariant is denoted by $\mathcal{D}^{coH}$.

Proposition 2.5**.**

Let $H$ be a finite dimensional Hopf algebra and let $\mathcal{D}$ be a small $K$-linear category. Then, $\mathcal{D}$ is a right co-$H$-category if and only if $\mathcal{D}$ is a left $H^{*}$-category. Moreover, $\mathcal{D}^{H^{*}}=\mathcal{D}^{coH}$.

Proof.

Let $\{e_{1},\dots,e_{n}\}$ be a basis of $H$ and let $\{e_{1}^{*},\dots,e_{n}^{*}\}$ be its dual basis. If $\mathcal{D}$ is a right co-$H$-category, then $\mathcal{D}$ becomes a left $H^{*}$-category with

[TABLE]

for all $h^{*}\in H^{*}$ and $f\in Hom_{\mathcal{D}}(X,Y)$. Indeed, it is easy to check that this action makes $Hom_{\mathcal{D}}(X,Y)$ a left $H^{*}$-module for every $X,Y\in Ob(\mathcal{D})$ and that

[TABLE]

Conversely, if $\mathcal{D}$ is a left $H^{*}$-category, then $\mathcal{D}$ is a right co-$H$-category with

[TABLE]

It may be verified that this gives a right $H$-comodule structure on $Hom_{\mathcal{D}}(X,Y)$. We need to check that the composition of morphisms in $\mathcal{D}$ is $H$-coequivariant. For any $h^{*}\in H^{*}$, $g\in Hom_{\mathcal{D}}(X,Y)$ and $f\in Hom_{\mathcal{D}}(Y,Z)$, we have

[TABLE]

Since $H$ is finite dimensional, it follows that

[TABLE]

We also have

[TABLE]

∎

Remark 2.6**.**

Using Example 2.3 and Proposition 2.5, we can obtain several examples of co-$H$-categories. Another example of a co-$H$-category is the smash extension $\mathcal{C}\#H$, which will be recalled in the next section.**

3 $H$-equivariant modules and the first spectral sequence

Let $\mathcal{C}$ be a left $H$-category. In this section, we will study the category of $H$-equivariant $\mathcal{C}$-modules and compute their higher derived $Hom$ functors by means of a spectral sequence. We begin with the following definition (see, for instance, [Sten, Mit2]).

Definition 3.1**.**

Let $\mathcal{C}$ be a small $K$-linear category. A right module over $\mathcal{C}$ is a $K$-linear functor $\mathcal{C}^{op}\longrightarrow Vect_{K}$, where $Vect_{K}$ denotes the category of $K$-vector spaces. Similarly, a left module over $\mathcal{C}$ is a $K$-linear functor $\mathcal{C}\longrightarrow Vect_{K}$. The category of all right (resp. left) modules over $\mathcal{C}$ will be denoted by $Mod$-$\mathcal{C}$ (resp. $\mathcal{C}$-$Mod$).

For each $X\in Ob(\mathcal{C})$, the representable functors ${\bf h}_{X}:=Hom_{\mathcal{C}}(-,X)$ and ${}_{X}{\bf h}:=Hom_{\mathcal{C}}(X,-)$ are examples of right and left modules over $\mathcal{C}$ respectively. Unless otherwise mentioned, by a $\mathcal{C}$-module we will always mean a right $\mathcal{C}$-module.

Definition 3.2**.**

Let $\mathcal{C}$ be a left $H$-category. Let $\mathcal{M}$ be a right $\mathcal{C}$-module with a given left $H$-module structure on $\mathcal{M}(X)$ for each $X\in Ob(\mathcal{C})$. Then, $\mathcal{M}$ is said to be a left $H$-equivariant right $\mathcal{C}$-module if

[TABLE]

A morphism $\eta:\mathcal{M}\longrightarrow\mathcal{N}$ of left $H$-equivariant right $\mathcal{C}$-modules is a morphism $\eta\in Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})$ such that $\eta(X):\mathcal{M}(X)\longrightarrow\mathcal{N}(X)$ is $H$-linear for each $X\in Ob(\mathcal{C})$. We will denote the category of left $H$-equivariant right $\mathcal{C}$-modules by $(Mod\text{-}\mathcal{C})_{H}^{H}$.

By $(Mod\text{-}\mathcal{C})_{H}$, we will denote the category whose objects are the same as those of $(Mod\text{-}\mathcal{C})_{H}^{H}$, but whose morphisms are those of right $\mathcal{C}$-modules.

Lemma 3.3**.**

Let $\mathcal{C}$ be a left $H$-category. Given $\mathcal{M},\mathcal{N}\in(Mod\text{-}\mathcal{C})_{H}$, the $H$-module action on $Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})$ given by

[TABLE]

for $\eta\in Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})$, $h\in H$, $X\in Ob(\mathcal{C})$, $m\in\mathcal{M}(X)$ makes $(Mod\text{-}\mathcal{C})_{H}$ a left $H$-category.

Proof.

Using the $H$-equivariance of $\mathcal{M}$ and $\mathcal{N}$, it may be verified that the action in (3.1) defines a left $H$-module structure on $Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})$. We now consider $\eta\in Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})$ and $\nu\in Hom_{Mod\text{-}\mathcal{C}}(\mathcal{N},\mathcal{P})$. Then, we have

[TABLE]

This proves the result. ∎

Proposition 3.4**.**

The category $(Mod\text{-}\mathcal{C})_{H}^{H}$ of left $H$-equivariant right $\mathcal{C}$-modules is identical to $((Mod\text{-}\mathcal{C})_{H})^{H}$.

Proof.

Suppose that $\eta\in Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})^{H}$. We claim that $\eta(X):\mathcal{M}(X)\longrightarrow\mathcal{N}(X)$ is $H$-linear for each $X\in Ob(\mathcal{C})$. For this, we observe that

[TABLE]

for any $h\in H$ and $m\in\mathcal{M}(X)$. Conversely, if each $\eta(X):\mathcal{M}(X)\longrightarrow\mathcal{N}(X)$ is $H$-linear, it is clear from the definition of the left $H$-action in (3.1) that $h\cdot\eta=\varepsilon(h)\eta$, i.e., $\eta\in Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})^{H}$. ∎

We will now study the category $(Mod\text{-}\mathcal{C})_{H}^{H}$ of left $H$-equivariant right $\mathcal{C}$-modules. In particular, one may ask if $(Mod\text{-}\mathcal{C})_{H}^{H}$ is an abelian category. We will show that $(Mod\text{-}\mathcal{C})_{H}^{H}$ is in fact a Grothendieck category. For this, we will need to consider the smash product category of $\mathcal{C}$ and $H$.

Definition 3.5**.**

(see [CiSo, $\S$ 2]) Let $\mathcal{C}$ be a left $H$-category. The smash product of $\mathcal{C}$ and $H$, denoted by $\mathcal{C}\#H$, is the $K$-linear category defined by

[TABLE]

An element of $Hom_{\mathcal{C}\#H}(X,Y)$ is a finite sum of the form $\sum g_{i}\#h_{i}$, with $g_{i}\in Hom_{\mathcal{C}}(X,Y)$ and $h_{i}\in H$. Then, the composition of morphisms in $\mathcal{C}\#H$ is determined by

[TABLE]

for any pair of composable morphisms $g:X\longrightarrow Y$, $f:Y\longrightarrow Z$ in $\mathcal{C}$ and any $h,h^{\prime}\in H$.

Lemma 3.6**.**

Let $\mathcal{M}\in Mod$-$(\mathcal{C}\#H)$. Then, $\mathcal{M}(X)$ has a left $H$-module structure for each $X\in Ob(\mathcal{C}\#H)$ given by

[TABLE]

Further, given any morphism $\eta:\mathcal{M}\longrightarrow\mathcal{N}$ in $Mod$-$(\mathcal{C}\#H)$, every $\eta(X):\mathcal{M}(X)\longrightarrow\mathcal{N}(X)$ is $H$-linear.

Proof.

For $h,h^{\prime}\in H$ and $m\in\mathcal{M}(X)$, we have

[TABLE]

If $\eta:\mathcal{M}\longrightarrow\mathcal{N}$ is a morphism in $Mod$-$(\mathcal{C}\#H)$, it may be verified easily that each $\eta(X):\mathcal{M}(X)\longrightarrow\mathcal{N}(X)$ is $H$-linear. ∎

Proposition 3.7**.**

Let $\mathcal{C}$ be a left $H$-category. Then, there is a one-one correspondence between left $H$-equivariant right $\mathcal{C}$-modules and right modules over $\mathcal{C}\#H$.

Proof.

For any $H$-equivariant $\mathcal{C}$-module $\mathcal{M}$, we have the object $\mathcal{M}^{\prime}$ in $Mod\text{-}(\mathcal{C}\#H)$ defined by

[TABLE]

For $f^{\prime}\#h^{\prime}\in Hom_{\mathcal{C}\#H}(Z,Y)$, $f\#h\in Hom_{\mathcal{C}\#H}(Y,X)$ and $m\in\mathcal{M}(X)$, we have

[TABLE]

Conversely, given any $\mathcal{M}^{\prime}$ in $Mod\text{-}(\mathcal{C}\#H)$, we can obtain an $H$-equivariant $\mathcal{C}$-module defined by

[TABLE]

From Lemma 3.6, it follows that $\mathcal{M}(X)=\mathcal{M}^{\prime}(X)$ has a left $H$-module structure. We now check that $\mathcal{M}$ is indeed $H$-equivariant:

[TABLE]

∎

Proposition 3.8**.**

Let $\mathcal{M}$ and $\mathcal{N}$ be right $\mathcal{C}\#H$-modules. Then, $Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})$ is a left $H$-module and its invariants are given by $Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})^{H}=Hom_{Mod\text{-}(\mathcal{C}\#H)}(\mathcal{M},\mathcal{N})$.

Proof.

We have shown in Proposition 3.7 that every right $\mathcal{C}\#H$-module is also a left $H$-equivariant right $\mathcal{C}$-module. Accordingly, we use (3.1) to give an $H$-module structure on $Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})$ by setting

[TABLE]

for any $\eta\in Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})$.

Suppose now that $\eta\in Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})^{H}$. From the proof of Proposition 3.4, it follows that $\eta(X):\mathcal{M}(X)\longrightarrow\mathcal{N}(X)$ is $H$-linear for each $X\in Ob(\mathcal{C})$. We need to show that $\eta\in Hom_{Mod\text{-}(\mathcal{C}\#H)}(\mathcal{M},\mathcal{N})$. For any $f:Y\longrightarrow X$ in $\mathcal{C}$, $h\in H$ and $m\in\mathcal{M}(X)$, we have

[TABLE]

Conversely, let $\eta\in Hom_{Mod\text{-}(\mathcal{C}\#H)}(\mathcal{M},\mathcal{N})$. Using the $H$-linearity of $\eta(X)$ from Lemma 3.6, it is clear from (3.3) that $\eta\in Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})^{H}$. ∎

Proposition 3.9**.**

Let $\mathcal{C}$ be a left $H$-category. Then, the categories $Mod\text{-}(\mathcal{C}\#H)$ and $(Mod\text{-}\mathcal{C})_{H}^{H}$ are identical. In particular, the category $(Mod\text{-}\mathcal{C})_{H}^{H}$ of left $H$-equivariant right $\mathcal{C}$-modules is a Grothendieck category.

Proof.

The fact that $Mod\text{-}(\mathcal{C}\#H)$ and $(Mod\text{-}\mathcal{C})_{H}^{H}$ are identical follows from Propositions 3.4, 3.7 and 3.8. Further, given any small preadditive category $\mathcal{E}$, it is well known that the category $Mod\text{-}\mathcal{E}$ is a Grothendieck category (see, for instance, [Sten, Example V.2.2]). Since $\mathcal{C}\#H$ is a small preadditive category, the result follows. ∎

We denote by $\mathcal{M}\otimes_{\mathcal{C}}(\mathcal{C}\#H)$ the extension of a right $\mathcal{C}$-module $\mathcal{M}$ to a right $(\mathcal{C}\#H)$-module. For the general notion of extension and restriction of scalars in the case of modules over a category, see, for instance, [DF, $\S$ 4]. It follows from [DF, Proposition 19] that the extension of scalars is left adjoint to the restriction of scalars.

Lemma 3.10**.**

Let $\mathcal{M}$ be a right $\mathcal{C}$-module. Then,

(1) A right $(\mathcal{C}\#H)$-module $\mathcal{M}\otimes H$ may be obtained by setting

[TABLE]

for any $X\in Ob(\mathcal{C}\#H)$, $f^{\prime}\#h^{\prime}\in Hom_{\mathcal{C}\#H}(Y,X)$, $m\in\mathcal{M}(X)$ and $h,h^{\prime}\in H$.

(2) $\mathcal{M}\otimes H$ is isomorphic to $\mathcal{M}\otimes_{\mathcal{C}}(\mathcal{C}\#H)$ as objects in $Mod\text{-}(\mathcal{C}\#H)$.

Proof.

(1) For any $f^{\prime\prime}\#h^{\prime\prime}\in Hom_{\mathcal{C}\#H}(Z,Y)$, $f^{\prime}\#h^{\prime}\in Hom_{\mathcal{C}\#H}(Y,X)$, we have

[TABLE]

Further, $\left((\mathcal{M}\otimes H)(id_{X}\#1_{H})\right)(m\otimes h)=\mathcal{M}(h_{1}id_{X})(m)\otimes h_{2}=m\otimes h$. Thus, $\mathcal{M}\otimes H\in Mod\text{-}(\mathcal{C}\#H)$.

(2) It may be easily checked that the assignment $\mathcal{M}\mapsto\mathcal{M}\otimes H$ defines a functor from $Mod$-$\mathcal{C}$ to $Mod$-$(\mathcal{C}\#H)$, which we denote by $({-})\otimes H$. We will now show that the functor $({-})\otimes H:Mod$-$\mathcal{C}\longrightarrow Mod$-$(\mathcal{C}\#H)$ is the left adjoint to the restriction of scalars from $Mod$-$(\mathcal{C}\#H)$ to $Mod$-$\mathcal{C}$, i.e., there is a natural isomorphism $Hom_{Mod\text{-}(\mathcal{C}\#H)}(\mathcal{M}\otimes H,\mathcal{N})\cong Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})$. The result of $(2)$ will then follow from the uniqueness of adjoints. We define

[TABLE]

by setting $\phi(\eta)(X)(m):=\eta(X)(m\otimes 1_{H})$ for any $\eta\in Hom_{Mod\text{-}(\mathcal{C}\#H)}(\mathcal{M}\otimes H,\mathcal{N})$, $X\in Ob(\mathcal{C})$ and $m\in\mathcal{M}(X)$. For any $f\in Hom_{\mathcal{C}}(X,Y)$ and $m^{\prime}\in\mathcal{M}(Y)$, we have

[TABLE]

Thus, $\phi(\eta)\in Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})$. We now check the injectivity of $\phi$. Let $\eta,\nu\in Hom_{Mod\text{-}(\mathcal{C}\#H)}(\mathcal{M}\otimes H,\mathcal{N})$ be such that $\phi(\eta)=\phi(\nu)$. Then, we have

[TABLE]

for all $X\in Ob(\mathcal{C})$ and $m\otimes h\in\mathcal{M}(X)\otimes H$. This shows that $\eta=\nu$. Next, given any $\xi\in Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})$, we define $\eta(X):\mathcal{M}(X)\otimes H\longrightarrow\mathcal{N}(X)$ by

[TABLE]

for $X\in Ob(\mathcal{C}\#H)$ and $m\otimes h\in\mathcal{M}(X)\otimes H$. Then, for any $f^{\prime}\#h^{\prime}\in Hom_{Mod\text{-}(\mathcal{C}\#H)}(Y,X)$, we have

[TABLE]

Hence, $\eta\in Hom_{Mod\text{-}(\mathcal{C}\#H)}(\mathcal{M}\otimes H,\mathcal{N})$. We also have $\phi(\eta)(X)(m)=\eta(X)(m\otimes 1_{H})=\xi(X)(m)$, i.e., $\phi(\eta)=\xi$. Hence, $\phi$ is surjective. This proves the result. ∎

Proposition 3.11**.**

(1) The extension of scalars from $Mod$-$\mathcal{C}$ to $Mod$-$(\mathcal{C}\#H)$ is exact.

(2) Let $\mathcal{I}$ be an injective object in $Mod\text{-}(\mathcal{C}\#H)$. Then, $\mathcal{I}$ is also an injective object in $Mod\text{-}\mathcal{C}$.

Proof.

Let $\mathcal{M},\mathcal{N}\in Mod$-$\mathcal{C}$ be such that $\phi:\mathcal{M}\longrightarrow\mathcal{N}$ is a monomorphism, i.e., $\phi(X):\mathcal{M}(X)\longrightarrow\mathcal{N}(X)$ is a monomorphism in $Vect_{K}$ for each $X\in Ob(\mathcal{C})$. Applying the isomorphism in Lemma 3.10, \big{(}\phi\otimes_{\mathcal{C}}(\mathcal{C}\#H)\big{)}(X)=(\phi\otimes H)(X):\mathcal{M}(X)\otimes H\longrightarrow\mathcal{N}(X)\otimes H is a monomorphism. Since extension of scalars is a left adjoint, it already preserves colimits. This proves (1). The result of (2) now follows from [SP, Tag 015Y]. ∎

Lemma 3.12**.**

Let $M\in H\text{-}Mod$ and let $\mathcal{N}\in Mod$-$(\mathcal{C}\#H)$. Then, a right $(\mathcal{C}\#H)$-module $M\otimes\mathcal{N}$ can be defined by setting

[TABLE]

for any $X\in\text{Ob}(\mathcal{C}),~{}f\in Hom_{\mathcal{C}}(Y,X)$ and $m\otimes n\in M\otimes\mathcal{N}(X)$.

Proof.

It is clear that $M\otimes\mathcal{N}\in Mod$-$\mathcal{C}$. Now for each $X\in Ob(\mathcal{C})$, the $K$-vector space $M\otimes\mathcal{N}(X)$ has a left $H$-module structure given by

[TABLE]

It may be easily verified that $M\otimes\mathcal{N}$ is an $H$-equivariant right $\mathcal{C}$-module under this action. Therefore, $M\otimes\mathcal{N}\in Mod$-$(\mathcal{C}\#H)$ by Proposition 3.7. ∎

Given any $\mathcal{N}\in Mod$-$(\mathcal{C}\#H)$, let $({-})\otimes\mathcal{N}:H\text{-}Mod\longrightarrow Mod$-$(\mathcal{C}\#H)$ denote the functor which takes any $M\in H$-$Mod$ to $M\otimes\mathcal{N}\in Mod$-$(\mathcal{C}\#H)$.

Proposition 3.13**.**

Let $\mathcal{N},\mathcal{P}\in Mod\text{-}(\mathcal{C}\#H)$ and let $M\in H\text{-}Mod$. Then, we have a natural isomorphism

[TABLE]

given by \big{(}\phi(\eta)(m)\big{)}(X)(n):=\eta(X)(m\otimes n) for each $X\in Ob(\mathcal{C})$ and $m\in M,n\in\mathcal{N}(X)$.

Proof.

Let $\eta\in Hom_{Mod\text{-}(\mathcal{C}\#H)}\left(M\otimes\mathcal{N},\mathcal{P}\right)$. It may be checked that $\phi(\eta)(m)\in Hom_{Mod\text{-}\mathcal{C}}(\mathcal{N},\mathcal{P})$ for every $m\in M$. We now verify that $\phi(\eta)$ is $H$-linear, i.e., for $h\in H$:

[TABLE]

Clearly, $\phi$ is injective. For f\in Hom_{H\text{-}Mod}\big{(}M,Hom_{Mod\text{-}\mathcal{C}}(\mathcal{N},\mathcal{P})\big{)}, we consider $\nu\in Hom_{Mod\text{-}(\mathcal{C}\#H)}(M\otimes\mathcal{N},\mathcal{P})$ determined by

[TABLE]

for each $X\in Ob(\mathcal{C}),n\in\mathcal{N}(X)$ and $m\in M$. We first check that $\nu(X):M\otimes\mathcal{N}(X)\longrightarrow\mathcal{P}(X)$ is $H$-linear for every $X\in Ob(\mathcal{C})$, i.e., for $h\in H$:

[TABLE]

Using the fact that $f(m)\in Hom_{Mod\text{-}\mathcal{C}}(\mathcal{N},\mathcal{P})$ for each $m\in M$, it may now be verified that $\nu\in Hom_{Mod\text{-}\mathcal{C}}(M\otimes\mathcal{N},\mathcal{P})$. From the equivalence of categories in Proposition 3.9, it follows that $\nu\in Hom_{Mod\text{-}(\mathcal{C}\#H)}\left(M\otimes\mathcal{N},\mathcal{P}\right)$. From (3.4), it is also clear that $\phi(\nu)=f$. ∎

Corollary 3.14**.**

If $\mathcal{I}$ is an injective object in $Mod$-$(\mathcal{C}\#H)$, then $Hom_{Mod\text{-}\mathcal{C}}(\mathcal{N},\mathcal{I})$ is an injective object in $H$-$Mod$ for any $\mathcal{N}\in Mod$-$(\mathcal{C}\#H)$.

Proof.

From Proposition 3.13, we know that the functor $(-)\otimes\mathcal{N}:H$-$Mod\longrightarrow Mod$-$(\mathcal{C}\#H)$ is a left adjoint and therefore preserves colimits. Further, given a monomorphism $M_{1}\hookrightarrow M_{2}$ in $H$-$Mod$, it is clear from the definition in Lemma 3.12 that $M_{1}\otimes\mathcal{N}\longrightarrow M_{2}\otimes\mathcal{N}$ is a monomorphism in $Mod$-$(\mathcal{C}\#H)$. Hence, $(-)\otimes\mathcal{N}:H$-$Mod\longrightarrow Mod$-$(\mathcal{C}\#H)$ is exact. As such, its right adjoint $Hom_{Mod\text{-}\mathcal{C}}(\mathcal{N},\_\_):Mod$-$(\mathcal{C}\#H)\longrightarrow H$-$Mod$ preserves injectives. ∎

We denote by $(-)^{H}$ the functor from $H\text{-}Mod$ to $Vect_{K}$ that takes $M$ to M^{H}=\{\mbox{m\in M$$|$$hm=\varepsilon(h)m$$\forall$$h\in H}\}. We now recall from Proposition 3.8 that we have an isomorphism

[TABLE]

for any $\mathcal{M}$, $\mathcal{N}\in Mod$-$(\mathcal{C}\#H)$. At the level of the derived $Hom$ functors, this leads to the following spectral sequence.

Theorem 3.15**.**

Let $\mathcal{M},~{}\mathcal{N}\in Mod\text{-}(\mathcal{C}\#H)$. Then, there exists a first quadrant spectral sequence:

[TABLE]

Proof.

We consider the functors $\mathscr{F}:=Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},-):Mod\text{-}(\mathcal{C}\#H)\longrightarrow H\text{-}{Mod}$ and $\mathscr{G}:=({-})^{H}:H{\text{-}}Mod\longrightarrow Vect_{K}$. We notice that $Mod$-$(\mathcal{C}\#H)$, $H$-$Mod$ and $Vect_{K}$ are all Grothendieck categories. From Corollary 3.14, we know that $\mathscr{F}$ preserves injectives. Using Proposition 3.8, we see that the functor $(\mathscr{G}\circ\mathscr{F}):Mod\text{-}(\mathcal{C}\#H)\longrightarrow Vect_{K}$ is given by $(\mathscr{G}\circ\mathscr{F})(\mathcal{N})=Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})^{H}=Hom_{Mod\text{-}(\mathcal{C}\#H)}(\mathcal{M},\mathcal{N})$. The result now follows from the Grothendieck spectral sequence for composite functors (see [Grothen]). ∎

4 $H$-locally finite modules and cohomology

We recall the definition of $H$-locally finite modules from [Gue]. For $M$ $\in H$-$Mod$ and $m\in M$, let $Hm$ be the $H$-submodule of $M$ spanned by the elements $hm$ for $h\in H$. Consider

[TABLE]

Clearly, $M^{(H)}$ is an $H$-submodule of $M$. An $H$-module $M$ is said to be $H$-locally finite if $M^{(H)}=M$. The full subcategory of $H$-$Mod$ whose objects are $H$-locally finite $H$-modules will be denoted by $H$-$mod$.

By Proposition 3.8, $Hom_{Mod\text{-}\mathcal{C}}(\mathcal{N},\mathcal{P})$ is an $H$-module for any $\mathcal{N},\mathcal{P}\in Mod\text{-}(\mathcal{C}\#H)$. We set

[TABLE]

Clearly, this defines a functor $\mathcal{L}_{Mod\text{-}\mathcal{C}}(\mathcal{N},{-}):Mod\text{-}(\mathcal{C}\#H)\longrightarrow H$-$mod$ for every $\mathcal{N}\in Mod\text{-}(\mathcal{C}\#H)$.

Proposition 4.1**.**

Let $\mathcal{N}\in Mod\text{-}(\mathcal{C}\#H)$. Then, the functor $\mathcal{L}_{Mod\text{-}\mathcal{C}}(\mathcal{N},{-}):Mod\text{-}(\mathcal{C}\#H)\longrightarrow H$-$mod$ is right adjoint to the functor $({-})\otimes\mathcal{N}:H$-$mod\longrightarrow Mod\text{-}(\mathcal{C}\#H)$, i.e., we have natural isomorphisms

[TABLE]

for all $\mathcal{P}\in Mod\text{-}(\mathcal{C}\#H)$ and $M\in H$-$mod$.

Proof.

Let \phi:Hom_{Mod\text{-}(\mathcal{C}\#H)}(M\otimes\mathcal{N},\mathcal{P})\longrightarrow Hom_{H\text{-}Mod}\big{(}M,Hom_{Mod\text{-}\mathcal{C}}(\mathcal{N},\mathcal{P})\big{)} be the isomorphism as in Proposition 3.13. Let $\eta:M\otimes\mathcal{N}\longrightarrow\mathcal{P}$ be a morphism in $Mod\text{-}(\mathcal{C}\#H)$. It follows that $H\phi(\eta)(m)$ is finite dimensional for each $m\in M$ by observing that $\phi(\eta)$ is $H$-linear and that $M$ is $H$-locally finite. Since $H$-$mod$ is a full subcategory of $H$-$Mod$, we have Hom_{Mod\text{-}(\mathcal{C}\#H)}(M\otimes\mathcal{N},\mathcal{P})\cong Hom_{H\text{-}Mod}\big{(}M,\mathcal{L}_{Mod\text{-}\mathcal{C}}(\mathcal{N},\mathcal{P})\big{)}\cong Hom_{H\text{-}mod}\big{(}M,\mathcal{L}_{Mod\text{-}\mathcal{C}}(\mathcal{N},\mathcal{P})\big{)}. ∎

For any $\mathcal{M}\in Mod$-$(\mathcal{C}\#H)$, we can now consider the functor

[TABLE]

Since $Mod\text{-}(\mathcal{C}\#H)$ is a Grothendieck category, we obtain derived functors $\mathbf{R}^{p}\mathcal{L}_{Mod\text{-}\mathcal{C}}(\mathcal{M},-):Mod\text{-}(\mathcal{C}\#H)\longrightarrow H\text{-}mod$, $p\geq 0$. We use the boldface notation to distinguish these from the functors ${R}^{p}\mathcal{L}_{Mod\text{-}\mathcal{C}}(\mathcal{M},-)$ that will appear later in the proof of Proposition 4.18 as derived functors of a restriction of $\mathcal{L}_{Mod\text{-}\mathcal{C}}(\mathcal{M},-)$.

Theorem 4.2**.**

Let $\mathcal{M},~{}\mathcal{N}\in Mod$-$(\mathcal{C}\#H)$. We consider the functors

[TABLE]

Then, we have the following spectral sequence

[TABLE]

Proof.

We have $(\mathscr{G}\circ\mathscr{F})(\mathcal{N})=Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})^{(H)}=\mathcal{L}_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})$. By definition,

[TABLE]

where $\mathcal{I}^{*}$ is an injective resolution of $\mathcal{N}$ in $Mod$-$(\mathcal{C}\#H)$. By Corollary 3.11, injectives in $Mod$-$(\mathcal{C}\#H)$ are also injectives in $Mod$-$\mathcal{C}$. Hence, $R^{q}\mathscr{F}(\mathcal{N})=\textnormal{Ext}_{Mod\text{-}\mathcal{C}}^{q}(\mathcal{M},\mathcal{N})$. For any injective $\mathcal{I}$ in $Mod$-$(\mathcal{C}\#H)$, we know that $\mathscr{F}(\mathcal{I})$ is injective in $H$-$Mod$ by Corollary 3.14. Since the category $H\text{-}Mod$ has enough injectives, the result now follows from Grothendieck spectral sequence for composite functors (see [Grothen]). ∎

Definition 4.3**.**

Let $\mathcal{C}$ be a left $H$-category.

• (1)

$\mathcal{C}$* is said to be $H$-locally finite if the $H$-module $Hom_{\mathcal{C}}(X,Y)$ is $H$-locally finite, i.e., $Hom_{\mathcal{C}}(X,Y)^{(H)}=Hom_{\mathcal{C}}(X,Y)$, for all $X,Y\in Ob(\mathcal{C})$.*

• (2)

Let $\mathcal{M}\in Mod\text{-}(\mathcal{C}\#H)$. Then, $\mathcal{M}$ is said to be $H$-locally finite if the $H$-module $\mathcal{M}(X)$ is $H$-locally finite, i.e. $\mathcal{M}(X)^{(H)}=\mathcal{M}(X)$, for each $X\in\text{Ob}(\mathcal{C})$. The full subcategory of $Mod$-$(\mathcal{C}\#H)$* whose objects are $H$-locally finite right $(\mathcal{C}\#H)$-modules will be denoted by $mod$-$(\mathcal{C}\#H)$.*

If $M$, $M^{\prime}\in H$-$Mod$, we know that $H$ acts diagonally on their tensor product $M\otimes M^{\prime}$ over $K$, i.e., $h(m\otimes m^{\prime})=\sum h_{1}m\otimes h_{2}m^{\prime}$ for $h\in H$, $m\in M$ and $m^{\prime}\in M^{\prime}$. In particular, if $M$, $M^{\prime}\in H$-$mod$, it follows that $M\otimes M^{\prime}\in H$-$mod$. Accordingly, if $\mathcal{N}\in mod\text{-}(\mathcal{C}\#H)$ and $M\in H$-$mod$, it is clear from the definition in Lemma 3.12 that $M\otimes\mathcal{N}\in mod\text{-}(\mathcal{C}\#H)$.

Corollary 4.4**.**

Let $\mathcal{N}\in mod\text{-}(\mathcal{C}\#H)$. Then, the functor $\mathcal{L}_{Mod\text{-}\mathcal{C}}(\mathcal{N},{-}):mod\text{-}(\mathcal{C}\#H)\longrightarrow H$-$mod$ is right adjoint to the functor $({-})\otimes\mathcal{N}:H$-$mod\longrightarrow mod\text{-}(\mathcal{C}\#H)$, i.e., we have natural isomorphisms

[TABLE]

for all $\mathcal{P}\in mod\text{-}(\mathcal{C}\#H)$ and $M\in H$-$mod$.

Proof.

This follows from Proposition 4.1 because $mod$-$(\mathcal{C}\#H)$ is a full subcategory of $Mod$-$(\mathcal{C}\#H)$. ∎

Lemma 4.5**.**

Let $\mathcal{C}$ be a left $H$-category. Given $X\in Ob(\mathcal{C})$, consider the representable functor ${\bf{h}}_{X}\in Mod$-$\mathcal{C}$. Then, the right $\mathcal{C}$-module ${\bf{h}}_{X}$ is also a right $(\mathcal{C}\#H)$-module.

Proof.

For each $Y\in Ob(\mathcal{C})$, we have ${\bf h}_{X}(Y)=Hom_{\mathcal{C}}(Y,X)$. Since $\mathcal{C}$ is a left $H$-category, ${\bf h}_{X}(Y)$ has a left $H$-module structure. For any $f\in Hom_{\mathcal{C}}(Z,Y)$, $g\in{\bf h}_{X}(Y)$ and $h\in H$, we have

[TABLE]

Thus, ${\bf h}_{X}$ is a left $H$-equivariant right $\mathcal{C}$-module. Hence, ${\bf h}_{X}\in Mod\text{-}(\mathcal{C}\#H)$ by Proposition 3.9. ∎

Lemma 4.6**.**

(1) If $\mathcal{I}$ is an injective in $Mod$-$(\mathcal{C}\#H)$, then $\mathcal{L}_{Mod\text{-}\mathcal{C}}(\mathcal{N},\mathcal{I})$ is an injective in $H$-$mod$ for any $\mathcal{N}\in Mod$-$(\mathcal{C}\#H)$.

(2) If $\mathcal{I}$ is an injective in $mod\text{-}(\mathcal{C}\#H)$, then

• (i)

$\mathcal{L}_{Mod\text{-}\mathcal{C}}(\mathcal{N},\mathcal{I})$* is an injective in $H$-$mod$ for any $\mathcal{N}\in mod\text{-}(\mathcal{C}\#H)$.*

• (ii)

Let $\mathcal{C}$ be $H$-locally finite. Then, for each $X\in\text{Ob}(\mathcal{C}\#H)$, $\mathcal{I}(X)$ is an injective in $H$-$mod$.

Proof.

(1) The functor $\mathcal{L}_{Mod\text{-}\mathcal{C}}(\mathcal{N},{-}):Mod\text{-}(\mathcal{C}\#H)\longrightarrow H$-$mod$ is right adjoint to the functor $(-)\otimes\mathcal{N}:H\text{-}mod\longrightarrow Mod\text{-}(\mathcal{C}\#H)$ by Proposition 4.1. Further, the functor $(-)\otimes\mathcal{N}$ always preserves monomorphisms (see the proof of Corollary 3.14). The result now follows from [SP, Tag 015Y].

(2) The proof of (i) is exactly the same as that of (1) except that we use Corollary 4.4 in place of Proposition 4.1. To prove (ii), we consider for each $X\in Ob(\mathcal{C})$ the representable functor ${\bf h}_{X}\in Mod$-$\mathcal{C}$. Using Lemma 4.5, we know that ${\bf{h}}_{X}\in Mod$-$(\mathcal{C}\#H)$. Further, since $\mathcal{C}$ is $H$-locally finite, we see that ${\bf{h}}_{X}\in mod$-$(\mathcal{C}\#H)$. Using (i), we have $\mathcal{L}_{Mod\text{-}\mathcal{C}}({\bf{h}}_{X},\mathcal{I})$ is injective in $H$-mod. Finally, by Yoneda lemma, we have $\mathcal{L}_{Mod\text{-}\mathcal{C}}({\bf{h}}_{X},\mathcal{I})=Hom_{Mod\text{-}\mathcal{C}}({\bf h}_{X},\mathcal{I})^{(H)}=\mathcal{I}(X)^{(H)}=\mathcal{I}(X)$. ∎

Lemma 4.7**.**

Let $\mathcal{C}$ be an $H$-locally finite category. Then, for any $\mathcal{M}$ in $Mod$-$(\mathcal{C}\#H)$, we may obtain an object $\mathcal{M}^{(H)}\in mod\text{-}(\mathcal{C}\#H)$ by setting

[TABLE]

for any $f\#h\in Hom_{\mathcal{C}\#H}(Y,X)$ and $m\in\mathcal{M}^{(H)}(X)$.

Proof.

We need to verify that $\mathcal{M}(f\#h):\mathcal{M}(X)\longrightarrow\mathcal{M}(Y)$ restricts to a map $\mathcal{M}(X)^{(H)}\longrightarrow\mathcal{M}(Y)^{(H)}$. For this, we consider $m\in\mathcal{M}(X)^{(H)}$. Since $\mathcal{M}\in Mod$-$(\mathcal{C}\#H)$ may be treated as a left $H$-equivariant right $\mathcal{C}$-module as in Proposition 3.7, we obtain

[TABLE]

for any $h^{\prime}\in H$. Since the category $\mathcal{C}$ is $H$-locally finite and $m\in\mathcal{M}^{(H)}(X)$, it is clear from (4.2) that $\mathcal{M}(f\#h)(m)\in\mathcal{M}(Y)^{(H)}=\mathcal{M}^{(H)}(Y)$. This proves the result. ∎

Proposition 4.8**.**

• (1)

Let $({-})^{(H)}:H{\text{-}}Mod\longrightarrow H{\text{-}}mod$ be the functor $N\mapsto N^{(H)}$. Then $({-})^{(H)}$ is right adjoint to the forgetful functor from the category $H{\text{-}}mod$ to the category $H{\text{-}}Mod$, i.e., we have natural isomorphisms

[TABLE]

for any $M\in H$-$mod$* and $N\in H$-$Mod$.*

• (2)

Let $({-})^{(H)}:Mod\text{-}(\mathcal{C}\#H)\longrightarrow mod\text{-}(\mathcal{C}\#H)$ be the functor $\mathcal{N}\mapsto\mathcal{N}^{(H)}$. Then $({-})^{(H)}$ is right adjoint to the forgetful functor from the category $mod\text{-}(\mathcal{C}\#H)$ to the category $Mod\text{-}(\mathcal{C}\#H)$, i.e., we have natural isomorphisms

[TABLE]

for any $\mathcal{M}\in mod$-$(\mathcal{C}\#H)$* and $\mathcal{N}\in Mod$-$(\mathcal{C}\#H)$.*

Proof.

(1) Given any $M\in H$-$mod$, $N\in H$-$Mod$ and an $H$-module morphism $\phi:M\longrightarrow N$, it is clear that $\phi(m)\in N^{(H)}$ for all $m\in M$.

(2) Let $\mathcal{M}\in mod\text{-}(\mathcal{C}\#H)$. By Lemma 3.6, a morphism $\eta:\mathcal{M}\longrightarrow\mathcal{N}$ in $Mod$-$(\mathcal{C}\#H)$ induces $H$-linear morphisms $\eta(X):\mathcal{M}(X)\longrightarrow\mathcal{N}(X)$ for each $X\in Ob(\mathcal{C})$. Since $\mathcal{M}(X)$ is $H$-locally finite, each $\eta(X)$ can be written as a morphism $\mathcal{M}(X)\longrightarrow\mathcal{N}(X)^{(H)}$. The result is now clear. ∎

Lemma 4.9**.**

Let $\mathcal{C}$ be $H$-locally finite. Then,

• (1)

The category $mod\text{-}(\mathcal{C}\#H)$ is abelian.

• (2)

If $\mathcal{I}$ is an injective in $Mod\text{-}(\mathcal{C}\#H)$, then $\mathcal{I}^{(H)}$ is an injective in $mod\text{-}(\mathcal{C}\#H)$.

• (3)

The category $mod\text{-}(\mathcal{C}\#H)$ has enough injectives.

Proof.

(1) Since $H$-$mod$ is closed under kernels and cokernels, it is clear that the subcategory $mod\text{-}(\mathcal{C}\#H)$ of the abelian category $Mod$-$(\mathcal{C}\#H)$ is closed under kernels and cokernels. Also, since products and coproducts of finitely many objects $\mathcal{M}_{i}$ in $mod\text{-}(\mathcal{C}\#H)$ are given by

[TABLE]

for $X\in Ob(\mathcal{C}\#H)$, it follows that finite products and coproducts exist and coincide in $mod\text{-}(\mathcal{C}\#H)$. Thus, the category $mod\text{-}(\mathcal{C}\#H)$ is abelian.

(2) Since the functor $({-})^{(H)}$ is right adjoint to the forgetful functor in Proposition 4.8(2) and the forgetful functor always preserves monomorphisms, this result follows from [SP, Tag 015Y].

(3) Since $Mod$-$(\mathcal{C}\#H)$ is a Grothendieck category, it has enough injectives. The result is now clear from (2). ∎

We now recall the notions of free, finitely generated and noetherian modules over a category from [Mit1, $\S$ 3] and [Mit2]. Given $\mathcal{M}\in Mod$-$\mathcal{C}$, we set $el(\mathcal{M}):=\underset{X\in Ob(\mathcal{C})}{\coprod}\mathcal{M}(X)$ to be the collection of all elements of $\mathcal{M}$. If $m\in el(\mathcal{M})$ lies in $\mathcal{M}(X)$, we write $|m|=X$.

Definition 4.10**.**

Let $\mathcal{C}$ be a small preadditive category and let $\mathcal{M}\in Mod$-$\mathcal{C}$.

• (i)

A family of elements $\{m_{i}\in el(\mathcal{M})\}_{i\in I}$ is said to generate $\mathcal{M}$ if every element $y\in el(\mathcal{M})$ can be expressed as $y=\sum_{i\in I}\mathcal{M}(f_{i})(m_{i})$ for some $f_{i}\in Hom_{\mathcal{C}}(|y|,|m_{i}|)$, where all but a finite number of the $f_{i}$ are zero. Equivalently, the family $\{m_{i}\in el(\mathcal{M})\}_{i\in I}$ is said to generate $\mathcal{M}$ if the induced morphism

[TABLE]

which takes $(0,...,0,id_{|m_{i}|},0,...,0)$ to $m_{i}$ is an epimorphism. The family is said to be a basis for $\mathcal{M}$ if $\eta$ is an isomorphism. The module $\mathcal{M}$ is said to be finitely generated (resp. free) if it has a finite set of generators (resp. a basis).

• (ii)

The module $\mathcal{M}$ is called noetherian if it satisfies the ascending chain condition on submodules. The category $\mathcal{C}$ is said to be right noetherian if ${\bf h}_{X}\in Mod\text{-}\mathcal{C}$ is noetherian for each $X\in Ob(\mathcal{C})$.

Proposition 4.11**.**

Let $\mathcal{C}$ be a left $H$-category. An object $\mathcal{M}\in mod$-$(\mathcal{C}\#H)$ is finitely generated in $Mod$-$(\mathcal{C}\#H)$ if and only if there exists a finite dimensional $V\in H$-$Mod$ and an epimorphism

[TABLE]

in $Mod$-$(\mathcal{C}\#H)$ for finitely many objects $\{X_{i}\}_{1\leq i\leq n}$ in $\mathcal{C}$, where each ${\bf h}_{X_{i}}$ is viewed as an object in $Mod$-$(\mathcal{C}\#H)$.

Proof.

Let $\mathcal{M}\in mod$-$(\mathcal{C}\#H)$ be finitely generated in $Mod$-$(\mathcal{C}\#H)$. We consider a finite generating family $\{m_{i}\in el(\mathcal{M})\}_{1\leq i\leq n}$ for $\mathcal{M}$. Since $\mathcal{M}\in mod$-$(\mathcal{C}\#H)$, each $\mathcal{M}(|m_{i}|)$ is $H$-locally finite and hence the $H$-module $V:=\underset{i=1}{\overset{n}{\bigoplus}}Hm_{i}$ is finite dimensional. For each $Y\in\text{Ob}(\mathcal{C}\#H)$, we consider the morphism determined by setting

[TABLE]

It is easy to check that $\eta$ is a morphism in $Mod$-$(\mathcal{C}\#H)$ and $\eta(Y)$ is an epimorphism for all $Y\in\text{Ob}(\mathcal{C}\#H)$. Conversely, let $\{v_{1},\ldots,v_{k}\}$ be a basis of a finite dimensional $H$-module $V$ and $X_{1},\ldots,X_{n}$ be finitely many objects in $\mathcal{C}$ such that there is an epimorphism

[TABLE]

in $Mod$-$(\mathcal{C}\#H)$. It may be verified that the elements $\{m_{ij}=\eta(X_{i})\left(v_{j}\otimes\text{id}_{X_{i}}\right)\in\mathcal{M}(X_{i})\}_{1\leq i\leq n,1\leq j\leq k}$ give a family of generators for $\mathcal{M}$. ∎

Corollary 4.12**.**

An object $\mathcal{M}$ in $mod$-$(\mathcal{C}\#H)$ is finitely generated in $Mod$-$(\mathcal{C}\#H)$ if and only if $\mathcal{M}$ is finitely generated in $Mod$-$\mathcal{C}$.

Proof.

Let $\mathcal{M}\in mod$-$(\mathcal{C}\#H)$ be finitely generated in $Mod$-$\mathcal{C}$. Then, there is a finite family $\{m_{i}\in el(\mathcal{M})\}_{i\in I}$ of elements of $\mathcal{M}$ such that every $y\in el(\mathcal{M})$ can be expressed as $y=\sum_{i\in I}\mathcal{M}(f_{i})(m_{i})$ for some $f_{i}\in Hom_{\mathcal{C}}(|y|,|m_{i}|)$. Then, $y=\sum_{i\in I}\mathcal{M}(f_{i}\#1_{H})(m_{i})$ and hence $\mathcal{M}$ is finitely generated as a $(\mathcal{C}\#H)$-module.

Conversely, let $\mathcal{M}$ in $mod$-$(\mathcal{C}\#H)$ be finitely generated in $Mod$-$(\mathcal{C}\#H)$ and let

[TABLE]

denote the epimorphism in $Mod$-$(\mathcal{C}\#H)$ as in Proposition 4.11. In particular, $\eta$ is an epimorphism in $Mod$-$\mathcal{C}$. Then, if $\{v_{1},...,v_{k}\}$ is a basis for $V$, it follows from the epimorphism in (4.3) that \{m_{ij}=\eta(X_{i})\big{(}v_{j}\otimes\text{id}_{X_{i}}\big{)}\in\mathcal{M}(X_{i})\}_{1\leq i\leq n,1\leq j\leq k} gives a finite set of generators for $\mathcal{M}$ as a right $\mathcal{C}$-module. ∎

We remark here that if $V$ and $V^{\prime}$ are left $H$-modules, then $Hom_{K}(V,V^{\prime})$ carries a left $H$-module action defined by

[TABLE]

This may be seen as the special case of the action described in Proposition 3.8 when $\mathcal{C}$ is the category with one object having endomorphism ring $K$.

Lemma 4.13**.**

Let $V\in H$-$Mod$ and $\mathcal{N}\in Mod$-$(\mathcal{C}\#H)$. Then, $Hom_{K}(V,\mathcal{N}(X))$ and Hom_{Mod\text{-}\mathcal{C}}\big{(}V\otimes{\bf{h}}_{X},\mathcal{N}\big{)} are isomorphic as objects in $H$-$Mod$ for each $X\in\text{Ob}(\mathcal{C})$.

Proof.

We check that the canonical isomorphism \phi:Hom_{Mod\text{-}\mathcal{C}}\big{(}V\otimes{\bf{h}}_{X},\mathcal{N}\big{)}\longrightarrow Hom_{K}(V,Hom_{Mod\text{-}\mathcal{C}}({\bf h}_{X},\mathcal{N}))\cong Hom_{K}(V,\mathcal{N}(X)) defined by $\phi(\eta)(v):=\eta(X)(v\otimes\text{id}_{X})$ for any morphism \eta\in Hom_{Mod\text{-}\mathcal{C}}\big{(}V\otimes{\bf{h}}_{X},\mathcal{N}\big{)} and $v\in V$ is $H$-linear:

[TABLE]

∎

Proposition 4.14**.**

Let $\mathcal{M}$ and $\mathcal{N}$ be in $mod$-$(\mathcal{C}\#H)$ with $\mathcal{M}$ finitely generated in $Mod$-$(\mathcal{C}\#H)$. Then, $Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})$ is $H$-locally finite, i.e., $\mathcal{L}_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})=Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})^{(H)}=Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})$.

Proof.

Since $\mathcal{M}\in mod$-$(\mathcal{C}\#H)$ is finitely generated in $Mod$-$(\mathcal{C}\#H)$, there exists by Proposition 4.11 a finite dimensional $H$-module $V$ and an epimorphism $\varphi:V\otimes\left(\bigoplus_{i=1}^{n}{\bf{h}}_{X_{i}}\right)\longrightarrow\mathcal{M}$ in $Mod$-$(\mathcal{C}\#H)$ for finitely many objects $X_{1},\ldots,X_{n}$ in $\mathcal{C}$. Thus we get a monomorphism

[TABLE]

For each $X\in\text{Ob}(\mathcal{C})$, $v\in V$ and $f\in{\bf{h}}_{X_{i}}(X)$ for some chosen $1\leq i\leq n$, we have

[TABLE]

This shows that $\hat{\varphi}$ is an $H$-module monomorphism. By Lemma 4.13, we know that

[TABLE]

as $H$-modules. Since $V$ is finite dimensional, we know that $Hom_{K}(V,\mathcal{N}(X_{i}))\cong\mathcal{N}(X_{i})\otimes V^{*}$ in $Vect_{K}$ and it is easily seen that this is an isomorphism of $H$-modules. Since $\mathcal{N}$ is an $H$-locally finite $(\mathcal{C}\#H)$-module and $V^{*}$ is $H$-locally finite (because $dim_{K}(V^{*})<\infty$), it follows that each $\mathcal{N}(X_{i})\otimes V^{*}$ is $H$-locally finite. The embedding in (4.5) now shows that $Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})$ is $H$-locally finite. ∎

Lemma 4.15**.**

Let $\mathcal{M}$, $\mathcal{N}\in Mod$-$(\mathcal{C}\#H)$. For a morphism $\eta\in Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})$, the following are equivalent:

(1) $\eta\in Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})^{(H)}=\mathcal{L}_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})$.

(2) There exists a finite dimensional $H$-module $V$, an element $v\in V$ and some $\hat{\eta}\in Hom_{Mod\text{-}(\mathcal{C}\#H)}(V\otimes\mathcal{M},\mathcal{N})$ such that $\hat{\eta}(X)(v\otimes m)=\eta(X)(m)$ for each $X\in Ob(\mathcal{C})$ and $m\in\mathcal{M}(X)$.

Proof.

(1) $\Rightarrow$ (2) : We put $V=H\eta$. Since $\eta\in Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})^{(H)}$, we see that $V$ is finite dimensional. Let $\{\eta_{1},...,\eta_{k}\}$ be a basis for $V=H\eta$. Any element $h\eta\in V$ can now be expressed as $h\eta=\sum_{i=1}^{k}\alpha_{i}(h)\eta_{i}$.

We now define $\hat{\eta}\in Hom_{Mod\text{-}\mathcal{C}}(V\otimes\mathcal{M},\mathcal{N})$ by setting $\hat{\eta}(X)(h\eta\otimes m):=(h\eta)(X)(m)$ for each $h\in H$, $X\in Ob(\mathcal{C})$ and $m\in\mathcal{M}(X)$. It is clear that $\hat{\eta}(X)(\eta\otimes m)=\eta(X)(m)$.

In order to show that $\hat{\eta}\in Hom_{Mod\text{-}(\mathcal{C}\#H)}(V\otimes\mathcal{M},\mathcal{N})$, it suffices to show that each $\hat{\eta}(X):V\otimes\mathcal{M}(X)\longrightarrow\mathcal{N}(X)$ is $H$-linear. For $h^{\prime}\in H$, we have

[TABLE]

(2) $\Rightarrow$ (1) : We are given $\hat{\eta}\in Hom_{Mod\text{-}(\mathcal{C}\#H)}(V\otimes\mathcal{M},\mathcal{N})$. Let $\{v_{1},...,v_{k}\}$ be a basis for $V$ and suppose that $hv=\sum_{i=1}^{k}\alpha_{i}(h)v_{i}$. For each $1\leq i\leq k$, we define $\xi_{i}\in Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})$ by setting $\xi_{i}(X)(m):=\hat{\eta}(X)(v_{i}\otimes m)$ for $X\in Ob(\mathcal{C})$, $m\in\mathcal{M}(X)$. For any $h\in H$, we see that

[TABLE]

It follows from the above that $H\eta$ lies in the space generated by the finite collection $\{\xi_{1},...,\xi_{k}\}\in Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})$. This proves the result. ∎

Proposition 4.16**.**

If $\mathcal{I}$ is an injective object in $mod$-$(\mathcal{C}\#H)$, then \mathcal{L}_{Mod\text{-}\mathcal{C}}\big{(}{-},\mathcal{I}\big{)} is an exact functor from $mod$-$(\mathcal{C}\#H)$ to $H$-$mod$.

Proof.

Let $0\longrightarrow\mathcal{M}\overset{i}{\longrightarrow}\mathcal{N}\longrightarrow\mathcal{P}\longrightarrow 0$ be an exact sequence in $mod$-$(\mathcal{C}\#H)$ and $\mathcal{I}$ be an injective object in $mod$-$(\mathcal{C}\#H)$. Then, $0\longrightarrow Hom_{Mod\text{-}\mathcal{C}}(\mathcal{P},\mathcal{I})\longrightarrow Hom_{Mod\text{-}\mathcal{C}}(\mathcal{N},\mathcal{I})\longrightarrow Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{I})$ is an exact sequence in $H$-$Mod$. Since the functor $(-)^{(H)}:H$-$Mod\longrightarrow H$-$mod$ is a right adjoint by Proposition 4.8(1), it preserves monomorphisms. Thus, $0\longrightarrow\mathcal{L}_{Mod\text{-}\mathcal{C}}(\mathcal{P},\mathcal{I})\longrightarrow\mathcal{L}_{Mod\text{-}\mathcal{C}}(\mathcal{N},\mathcal{I})\longrightarrow\mathcal{L}_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{I})$ is an exact sequence in $H$-$mod$.

Let $\eta\in\mathcal{L}_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{I})$. We set $V:=H\eta$. Then, $V$ is a finite dimensional $H$-module and therefore $V\in H$-$mod$. Thus, $V\otimes\mathcal{M},V\otimes\mathcal{N}\in mod$-$(\mathcal{C}\#H)$ and we have a monomorphism ${id}_{V}\otimes i:V\otimes\mathcal{M}\longrightarrow V\otimes\mathcal{N}$ in $mod$-$(\mathcal{C}\#H)$. Now, we consider the morphism $\zeta\in Hom_{Mod\text{-}\mathcal{C}}(V\otimes\mathcal{M},\mathcal{I})$ defined by setting $\zeta(X)(\nu\otimes m):=\nu(X)(m)$ for each $X\in Ob(\mathcal{C})$, $m\in\mathcal{M}(X)$ and $\nu\in V$. It may be verified that $\zeta(X)$ is $H$-linear for each $X\in Ob(\mathcal{C})$. Thus, $\zeta\in Hom_{mod\text{-}(\mathcal{C}\#H)}(V\otimes\mathcal{M},\mathcal{I})$. Since $\mathcal{I}$ is injective in $mod$-$(\mathcal{C}\#H)$, there exists a morphism $\xi:V\otimes\mathcal{N}\longrightarrow\mathcal{I}$ in $mod$-$(\mathcal{C}\#H)$ such that $\xi(\text{id}_{V}\otimes i)=\zeta$. The morphism $\xi\in Hom_{Mod\text{-}(\mathcal{C}\#H)}(V\otimes\mathcal{N},\mathcal{I})$ now induces a morphism $\hat{\xi}\in Hom_{Mod\text{-}\mathcal{C}}(\mathcal{N},\mathcal{I})$ defined by setting $\hat{\xi}(X)(n):=\xi(X)(\eta\otimes n)$ for every $X\in Ob(\mathcal{C})$ and $n\in\mathcal{N}(X)$. Applying Lemma 4.15, we see that $\hat{\xi}\in\mathcal{L}_{Mod\text{-}\mathcal{C}}(\mathcal{N},\mathcal{I})$. Also, $\hat{\xi}\circ i=\eta$. This completes the proof. ∎

Proposition 4.17**.**

Let $\mathcal{C}$ be a left $H$-locally finite category which is right noetherian. Let $\mathcal{M}\in mod\text{-}(\mathcal{C}\#H)$ be finitely generated as an object in $Mod$-$(\mathcal{C}\#H)$. If $\mathcal{I}$ is an injective object in $mod$-$(\mathcal{C}\#H)$, then $\textnormal{Ext}_{Mod\text{-}\mathcal{C}}^{p}(\mathcal{M},\mathcal{I})=0$ for all $p>0$.

Proof.

Since $\mathcal{M}\in mod$-$(\mathcal{C}\#H)$ is finitely generated in $Mod$-$(\mathcal{C}\#H)$, by Proposition 4.11, there exists a finite dimensional $H$-module $V_{0}$ and an epimorphism

[TABLE]

in $Mod$-$(\mathcal{C}\#H)$ for finitely many objects $\{X_{i}\}_{1\leq i\leq n_{0}}$ in $\mathcal{C}$, where ${\bf{h}}_{X_{i}}$ are viewed as objects in $Mod$-$(\mathcal{C}\#H)$. Since $\mathcal{C}$ is $H$-locally finite, each ${\bf{h}}_{X_{i}}\in mod$-$(\mathcal{C}\#H)$. Since $V_{0}$ is finite dimensional, we must have $V_{0}\in H$-$mod$. Thus, $\mathcal{P}_{0}\in mod$-$(\mathcal{C}\#H)$. Using Proposition 4.11 and Corollary 4.12, it follows that $\mathcal{P}_{0}$ is finitely generated in $Mod$-$\mathcal{C}$. Since $\mathcal{C}$ is right noetherian, $\mathcal{P}_{0}$ is a noetherian right $\mathcal{C}$-module (see, for instance, [Mit1, $\S$ 3]). Since the submodule of a finitely generated noetherian module is finitely generated, the $(\mathcal{C}\#H)$-submodule $\mathcal{K}:=Ker(\eta_{0})$ of $\mathcal{P}_{0}$ is finitely generated in $Mod$-$\mathcal{C}$. So, again using Proposition 4.11 and Corollary 4.12 it follows that there exists a finite dimensional $H$-module $V_{1}$ and an epimorphism

[TABLE]

in $Mod$-$(\mathcal{C}\#H)$ for finitely many objects $\{Y_{j}\}_{1\leq j\leq n_{1}}$ in $\mathcal{C}$. Since $V_{0}$ and $V_{1}$ are finite dimensional $K$-vector spaces, clearly $\mathcal{P}_{0}$ and $\mathcal{P}_{1}$ are free right $\mathcal{C}$-modules. Moreover, $Im(\eta_{1})=\mathcal{K}=Ker(\eta_{0})$. Thus, continuing in this way, we can construct a free resolution of the module $\mathcal{M}$ in the category $Mod$-$\mathcal{C}$:

[TABLE]

Hence, we have

[TABLE]

Since $\mathcal{M}$ and $\{\mathcal{P}_{i}\}_{i\geq 0}$ are finitely generated in $Mod\text{-}(\mathcal{C}\#H)$, we have $\mathcal{L}_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{I})=Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{I})$ and $\mathcal{L}_{Mod\text{-}\mathcal{C}}(\mathcal{P}_{i},\mathcal{I})=Hom_{Mod\text{-}\mathcal{C}}(\mathcal{P}_{i},\mathcal{I})$ for every $i\geq 0$, by Proposition 4.14. From Proposition 4.16, we know that $\mathcal{L}_{Mod\text{-}\mathcal{C}}(-,\mathcal{I})$ is exact and it follows that $H^{p}(\mathcal{L}_{Mod\text{-}\mathcal{C}}(\mathcal{P}_{*},\mathcal{I}))=0$ for all $p>0$. This proves the result. ∎

Proposition 4.18**.**

Let $\mathcal{C}$ be a left $H$-locally finite category which is right noetherian. Let $\mathcal{M}$ and $\mathcal{N}$ be in $mod$-$(\mathcal{C}\#H)$ with $\mathcal{M}$ finitely generated in $Mod$-$(\mathcal{C}\#H)$ and let $\mathcal{E}^{*}$ be an injective resolution of $\mathcal{N}$ in $mod$-$(\mathcal{C}\#H)$. Then,

[TABLE]

Proof.

Let $\mathcal{P}_{*}$ be the free resolution of $\mathcal{M}$ in $Mod$-$\mathcal{C}$ constructed as in the proof of Proposition 4.17. Then, we have

[TABLE]

where the second equality follows from Proposition 4.14. Since $H$-$mod$ is an abelian category and $\mathcal{L}_{Mod\text{-}\mathcal{C}}(\mathcal{P}_{*},\mathcal{N})$ is a complex in $H$-$mod$, it follows that H^{p}\big{(}\mathcal{L}_{Mod\text{-}\mathcal{C}}(\mathcal{P}_{*},\mathcal{N})\big{)}\in H-$mod$. Hence, we may consider the family $\{\textnormal{Ext}_{Mod\text{-}\mathcal{C}}^{p}(\mathcal{M},-)\}_{p\geq 0}$ as a $\delta$-functor from $mod$-$(\mathcal{C}\#H)$ to $H$-$mod$.

By Proposition 4.17, $\textnormal{Ext}_{Mod\text{-}\mathcal{C}}^{p}(\mathcal{M},\mathcal{I})=0,~{}p>0$ for every injective object $\mathcal{I}$ in $mod\text{-}(\mathcal{C}\#H)$. Since $mod$-$(\mathcal{C}\#H)$ has enough injectives, it follows that each $\textnormal{Ext}_{Mod\text{-}\mathcal{C}}^{p}(\mathcal{M},-):mod$-$(\mathcal{C}\#H)\longrightarrow H$-$mod$ is effaceable (see, for instance, [Hart, $\S$ III.1]).

Since $mod$-$(\mathcal{C}\#H)$ has enough injectives, we can consider the right derived functors

[TABLE]

For $p=0$, we notice that $\textnormal{Ext}_{Mod\text{-}\mathcal{C}}^{0}(\mathcal{M},-)=Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},-)=\mathcal{L}_{Mod\text{-}\mathcal{C}}(\mathcal{M},-)=R^{0}\mathcal{L}_{Mod\text{-}\mathcal{C}}(\mathcal{M},-)$ as functors from $mod\text{-}(\mathcal{C}\#H)$ to $H$-$mod$. Since each $\textnormal{Ext}_{Mod\text{-}\mathcal{C}}^{p}(\mathcal{M},-)$ is effaceable for $p>0$, the family $\{\textnormal{Ext}_{Mod\text{-}\mathcal{C}}^{p}(\mathcal{M},-)\}_{p\geq 0}$ forms a universal $\delta$-functor and it follows from [Hart, Corollary III.1.4] that

[TABLE]

for every $p\geq 0$. Therefore, we have

[TABLE]

∎

Theorem 4.19**.**

Let $\mathcal{C}$ be a left $H$-locally finite category which is right noetherian. Fix $\mathcal{M}\in mod$-$(\mathcal{C}\#H)$ with $\mathcal{M}$ finitely generated in $Mod$-$(\mathcal{C}\#H)$. We consider the functors

[TABLE]

Then, we have the following spectral sequence

[TABLE]

Proof.

Using Proposition 3.8 and the fact that $mod$-$(\mathcal{C}\#H)$ is a full subcategory of $Mod$-$(\mathcal{C}\#H)$, we have $(\mathscr{G}\circ\mathscr{F})(\mathcal{N})=Hom_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})^{H}=Hom_{mod\text{-}(\mathcal{C}\#H)}(\mathcal{M},\mathcal{N})$. By definition,

[TABLE]

where $\mathcal{E}^{*}$ is an injective resolution of $\mathcal{N}$ in $mod$-$(\mathcal{C}\#H)$. By Proposition 4.18, we get $R^{q}\mathscr{F}(\mathcal{N})=\textnormal{Ext}^{q}_{Mod\text{-}\mathcal{C}}(\mathcal{M},\mathcal{N})$. For any injective $\mathcal{I}$ in $mod$-$(\mathcal{C}\#H)$, we know that $\mathscr{F}(\mathcal{I})$ is an injective in $H$-$mod$ by Proposition 4.14 and Lemma 4.6(2). Since the category $H\text{-}mod$ has enough injectives (see,[Gue, Lemma 1.4]), the result now follows from Grothendieck spectral sequence for composite functors (see [Grothen]). ∎

5 Cohomology of relative $(\mathcal{D},H)$-Hopf modules

Let $\mathcal{D}$ be a right co-$H$-category. In the notation of Definition 2.4, for any $X$, $Y\in Ob(\mathcal{D})$, there is an $H$-coaction on the $K$-vector space $Hom_{\mathcal{D}}(X,Y)$ given by $\rho_{XY}(f):=\sum f_{0}\otimes f_{1}$. In this section, we will study the relative Hopf modules over the category $\mathcal{D}$ and describe their derived $Hom$-functors by means of spectral sequences.

We denote by $Comod$-$H$ the category of right $H$-comodules. If $M$ is an $H$-comodule with right $H$-coaction given by $\rho_{M}:M\longrightarrow M\otimes H$, we set $M^{coH}:=\{m\in M~{}|~{}\rho_{M}(m)=m\otimes 1_{H}\}$ to be the coinvariants of $M$.

Definition 5.1**.**

Let $\mathcal{D}$ be a right co-$H$-category. Let $\mathcal{M}$ be a left $\mathcal{D}$-module with a given right $H$-comodule structure $\rho_{\mathcal{M}(X)}:\mathcal{M}(X)\longrightarrow\mathcal{M}(X)\otimes H$, $m\mapsto\sum m_{0}\otimes m_{1}$ on $\mathcal{M}(X)$ for each $X$ in $Ob(\mathcal{D})$. Then, $\mathcal{M}$ is said to be a relative $(\mathcal{D},H)$-Hopf module if the following condition holds:

[TABLE]

for any $f\in Hom_{\mathcal{D}}(X,Y)$ and $m\in\mathcal{M}(X)$. We denote by ${}_{\mathcal{D}}{\mathscr{M}}^{H}$ the category whose objects are relative $(\mathcal{D},H)$-Hopf modules and whose morphisms are given by

[TABLE]

We now recall the tensor product of $H$-comodules. Let $M,N\in Comod$-$H$ with $H$-coactions $\rho_{M}$ and $\rho_{N}$, respectively. Then, $M\otimes N\in Comod$-$H$ with $H$-coaction given by $\rho_{M\otimes N}:=(id\otimes id\otimes m_{H})(id\otimes\tau\otimes id)(\rho_{M}\otimes\rho_{N})$, where $m_{H}$ denotes the multiplication on $H$ and $id\otimes\tau\otimes id:M\otimes(H\otimes N)\otimes H\longrightarrow M\otimes(N\otimes H)\otimes H$ denotes the twist map. In other words, $\rho_{M\otimes N}(m\otimes n)=\sum m_{0}\otimes n_{0}\otimes m_{1}n_{1}$ for $m\otimes n\in M\otimes N$.

Lemma 5.2**.**

Let $M\in Comod\text{-}H$ and $\mathcal{N}\in{{}_{\mathcal{D}}}{\mathscr{M}}^{H}$. Then, $\mathcal{N}\otimes M$ defined by setting

[TABLE]

for each $X\in\text{Ob}(\mathcal{D}),~{}f\in Hom_{\mathcal{D}}(X,Y)$ and $n\otimes m\in\mathcal{N}(X)\otimes M$ is a relative $(\mathcal{D},H)$-Hopf module.

Proof.

Clearly, $\mathcal{N}\otimes M$ is a left $\mathcal{D}$-module. Since $\mathcal{N}(X)$ is a right $H$-comodule, $\mathcal{N}(X)\otimes M$ also carries a right $H$-comodule structure for each $X\in\text{Ob}(\mathcal{D})$. For any $f\in Hom_{\mathcal{D}}(X,Y)$ and $n\otimes m\in\mathcal{N}(X)\otimes M$, we have

[TABLE]

This shows that $\mathcal{N}\otimes M$ satisfies the condition (5.1) in Definition 5.1. ∎

From Lemma 5.2, it follows that the assignment $M\mapsto\mathcal{N}\otimes M$ defines a functor $\mathcal{N}\otimes(-):Comod\text{-}H\longrightarrow{{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ for each $\mathcal{N}\in{{}_{\mathcal{D}}}{\mathscr{M}}^{H}$.

From the definition of a co-$H$-category, it is also clear that the $\mathcal{D}$-module ${{}_{X}}{\bf{h}}:=Hom_{\mathcal{D}}(X,{-})$ lies in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ for each $X\in Ob(\mathcal{D})$.

Lemma 5.3**.**

Let $\mathcal{M}$ be a relative $(\mathcal{D},H)$-Hopf module and let $m\in\mathcal{M}(X)$ for some $X\in Ob(\mathcal{D})$. Then, there exists a finite dimensional $H$-comodule $W_{m}\subseteq\mathcal{M}(X)$ containing $m$ and a morphism $\eta_{m}:{{}_{X}}{\bf{h}}\otimes W_{m}\longrightarrow\mathcal{M}$ in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ such that $\eta_{m}(X)(id_{X}\otimes m)=m$.

Proof.

Using [SCS, Theorem 2.1.7], we know that there exists a finite dimensional $H$-subcomodule $W_{m}$ of $\mathcal{M}(X)$ containing $m$. We consider the $\mathcal{D}$-module morphism $\eta_{m}:{{}_{X}}{\bf{h}}\otimes W_{m}\longrightarrow\mathcal{M}$ defined by

[TABLE]

for any $Y\in Ob(\mathcal{D})$, $f\in{{}_{X}}{\bf{h}}(Y)$ and $w\in W_{m}$. We now verify that $\eta_{m}$ is indeed a morphism in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$, i.e., $\eta_{m}(Y)$ is $H$-colinear for each $Y\in Ob(\mathcal{D})$:

[TABLE]

∎

Remark 5.4**.**

*It might be tempting to view Lemma 5.3 as a Yoneda correspondence. But, we note that the finite dimensional $H$-comodule and the morphism in Lemma 5.3 determined by $m\in\mathcal{M}(X)$ need not be unique. ***

Given a morphism $\eta:\mathcal{M}\longrightarrow\mathcal{N}$ in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$, it may be easily verified that $Ker(\eta)$ and $Coker(\eta)$ determined by setting

[TABLE]

for each $X\in Ob(\mathcal{D})$ are also relative $(\mathcal{D},H)$-Hopf modules. It follows that $\eta:\mathcal{M}\longrightarrow\mathcal{N}$ in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ is a monomorphism (resp. an epimorphism) if and only if it induces monomorphisms (resp. epimorphisms) $\eta(X):\mathcal{M}(X)\longrightarrow\mathcal{N}(X)$ of $H$-comodules for each $X\in Ob(\mathcal{D})$.

Proposition 5.5**.**

Let $\mathcal{D}$ be a right co-$H$-category. Then, a module $\mathcal{M}\in{{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ is finitely generated as an object in $\mathcal{D}\text{-}Mod$ if and only if there exists a finite dimensional $H$-comodule $W$ and an epimorphism

[TABLE]

in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$, for finitely many objects $\{X_{i}\}_{i\in I}$ in $\mathcal{D}$.

Proof.

Let $\mathcal{M}\in{{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ be finitely generated as a $\mathcal{D}$-module. Then, there exists a finite collection $\{m_{i}\in el(\mathcal{M})\}_{i\in I}$ such that every $y\in el(\mathcal{M})$ has the form $y=\sum_{i\in I}\mathcal{M}(f_{i})(m_{i})$ for some $f_{i}\in Hom_{\mathcal{D}}(|m_{i}|,|y|)$. Applying Lemma 5.3, we can obtain for each $m_{i}$ a finite dimensional $H$-subcomodule $W_{m_{i}}\subseteq\mathcal{M}(|m_{i}|)$ containing $m_{i}$ and a morphism $\eta_{m_{i}}:{{}_{|m_{i}|}}{\bf{h}}\otimes W_{m_{i}}\longrightarrow\mathcal{M}$ in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$. Setting $W:=\bigoplus_{i\in I}W_{m_{i}}$, we have an epimorphism in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ determined by

[TABLE]

for each $Y\in Ob(\mathcal{D})$, $f_{i}\in{{}_{|m_{i}|}}{\bf{h}}(Y)$ and $w_{i}\in W_{m_{i}}$.

Conversely, let $\{w_{1},\ldots,w_{k}\}$ be a basis of a finite dimensional $H$-comodule $W$ and $\{X_{i}\}_{i\in I}$ be finitely many objects in $\mathcal{D}$ such that we have an epimorphism

[TABLE]

in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$. From the discussion above, it follows that $\eta(Y):\left(\bigoplus_{i\in I}{{}_{X_{i}}}{\bf{h}}(Y)\right)\otimes W\longrightarrow\mathcal{M}(Y)$ is an epimorphism in $Comod$-$H$ for each $Y\in Ob(\mathcal{D})$. Then, the elements \{m_{ij}:=\eta(X_{i})\big{(}\text{id}_{X_{i}}\otimes w_{j}\big{)}\}_{i\in I,1\leq j\leq k} form a family of generators for $\mathcal{M}$ as a $\mathcal{D}$-module. ∎

We will now show that ${}_{\mathcal{D}}{\mathscr{M}}^{H}$ is a Grothendieck category. This essentially follows from the fact that both $\mathcal{D}$-$Mod$ and $Comod$-$H$ are Grothendieck categories. We refer the reader, for instance, to [SCS, Corollary 2.2.8], for a proof of $Comod$-$H$ being a Grothendieck category.

Proposition 5.6**.**

Let $\mathcal{D}$ be a right co-$H$-category. Then, the category ${}_{\mathcal{D}}{\mathscr{M}}^{H}$ of relative $(\mathcal{D},H)$-Hopf modules is a Grothendieck category.

Proof.

Since the categories $\mathcal{D}$-$Mod$ and $Comod$-$H$ have kernels, cokernels and coproducts (direct sums), so does the category ${}_{\mathcal{D}}{\mathscr{M}}^{H}$. The remaining properties of an abelian category are inherited by ${}_{\mathcal{D}}{\mathscr{M}}^{H}$ from $\mathcal{D}$-$Mod$. Hence, ${}_{\mathcal{D}}{\mathscr{M}}^{H}$ is a cocomplete abelian category. Directs limits are exact in ${}_{\mathcal{D}}{\mathscr{M}}^{H}$ which is also a property inherited from $\mathcal{D}$-$Mod$. We are now left to check that ${}_{\mathcal{D}}{\mathscr{M}}^{H}$ has a family of generators. For any $\mathcal{M}$ in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$, it follows from Lemma 5.3 that we can find an epimorphism

[TABLE]

in ${}_{\mathcal{D}}{\mathscr{M}}^{H}$. Thus, the collection $\{_{X}{\bf h}\otimes W\}$, where $X$ ranges over all objects in $\mathcal{D}$ and $W$ ranges over all (isomorphism classes of) finite dimensional $H$-comodules, forms a generating family for ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ (see, for instance, the proof of [Grothen, Proposition 1.9.1]). ∎

For $\mathcal{N}\in{{}_{\mathcal{D}}}{\mathscr{M}}^{H}$, we consider the functor $\mathcal{N}\otimes{(-)}:Comod\textnormal{-}H$ $\longrightarrow{{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ given by $M\mapsto\mathcal{N}\otimes M$. We see that $Comod\textnormal{-}H$ is a Grothendieck category and the functor $\mathcal{N}\otimes{(-)}$ preserves colimits. Therefore, by a classical result [KS, Proposition 8.3.27(iii)], it has a right adjoint which we denote by $\mathscr{R}_{\mathcal{N}}:{{}_{\mathcal{D}}}{\mathscr{M}}^{H}\longrightarrow Comod\textnormal{-}H$. We then define

[TABLE]

for any $\mathcal{P}\in{{}_{\mathcal{D}}}{\mathscr{M}}^{H}$. Thus, we have a natural isomorphism

[TABLE]

for $\mathcal{N},\mathcal{P}\in{{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ and $M\in Comod$-$H$. In particular, when $\mathcal{D}$ is a right co-$H$-category with a single object, i.e., a right $H$-comodule algebra, then $\mathcal{N}$ and $\mathcal{P}$ are relative Hopf-modules in the classical sense of Takeuchi [t]. Then, using [CanGue, Lemma 2.3], the definition of $HOM$ as in (5.3) recovers the standard definition of rational morphisms between relative Hopf modules as in [CanGue, $\S$ 2] or [Ulb]. As such, we will refer to $HOM_{\mathcal{D}\text{-}Mod}(-,-)$ as the “rational $Hom$ object” in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$.

Corollary 5.7**.**

Let $\mathcal{N},\mathcal{P}\in{{}_{\mathcal{D}}}{\mathscr{M}}^{H}$. Then, $HOM_{\mathcal{D}\text{-}Mod}(\mathcal{N},\mathcal{P})^{coH}=Hom_{{{}_{\mathcal{D}}}{\mathscr{M}}^{H}}(\mathcal{N},\mathcal{P})$.

Proof.

The result follows by choosing $M=k$ in (5.4) and the fact that $Hom_{Comod\text{-}H}(k,N)=N^{coH}$ for any $N\in Comod$-$H$. ∎

Corollary 5.8**.**

If $\mathcal{I}$ is an injective in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$, then $HOM_{\mathcal{D}\text{-}Mod}(\mathcal{N},\mathcal{I})$ is an injective in $Comod$-$H$ for any $\mathcal{N}$ in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$.

Proof.

The fact that $HOM_{\mathcal{D}\text{-}Mod}(\mathcal{N},-):{{}_{\mathcal{D}}}{\mathscr{M}}^{H}\longrightarrow Comod\text{-}H$ preserves injectives follows from the fact that its left adjoint $\mathcal{N}\otimes(-):Comod\text{-}H\longrightarrow{{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ is an exact functor. ∎

At the level of higher derived functors, the result of Corollary 5.7 leads to the following spectral sequence.

Theorem 5.9**.**

Let $\mathcal{M}\in{{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ be a relative $(\mathcal{D},H)$-Hopf module. We consider the functors

[TABLE]

Then, we have the following spectral sequence

[TABLE]

Proof.

We have $(\mathscr{G}\circ\mathscr{F})(\mathcal{N})=HOM_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{N})^{coH}=Hom_{{{}_{\mathcal{D}}}{\mathscr{M}}^{H}}(\mathcal{M},\mathcal{N})$ by Corollary 5.7. By Corollary 5.8, the functor $\mathscr{F}$ preserves injectives. Since $Comod$-$H$ has enough injectives, the result now follows from Grothendieck spectral sequence for composite functors (see [Grothen]). ∎

Let $M,N$ be right $H$-comodules. Let $H^{*}$ be the linear dual of $H$. Then, the space $Hom_{K}(M,N)$ carries a left $H^{*}$-module structure given by

[TABLE]

for any $h^{*}\in H^{*}$, $f\in Hom_{K}(M,N)$ and $m\in M$. We now show that this $H^{*}$-action can be extended to relative $(\mathcal{D}\text{-}H)$-Hopf modules.

Lemma 5.10**.**

Let $\mathcal{M},~{}\mathcal{N}\in{{}_{\mathcal{D}}}{\mathscr{M}}^{H}$. Then, $Hom_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{N})$ is a left $H^{*}$-module.

Proof.

For $h^{*}\in H^{*}$ and $\eta\in Hom_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{N})$, we set

[TABLE]

for all $X\in Ob(\mathcal{D})$ and $m\in\mathcal{M}(X)$. We first verify that $h^{*}\eta$ is indeed an element in $Hom_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{N})$. For any $f\in Hom_{\mathcal{D}}(X,Y)$, we have

[TABLE]

Next, we verify that $(h^{*}g^{*})\eta=h^{*}(g^{*}\eta)$ and that $1_{H^{*}}\eta=\eta$, i.e., $\varepsilon\eta=\eta$ for all $h^{*},g^{*}\in H^{*}$ and $\eta\in Hom_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{N})$. The latter equality follows easily and further we see that

[TABLE]

for all $X\in Ob(\mathcal{D})$ and $m\in\mathcal{M}(X)$. ∎

Lemma 5.11**.**

Let $\mathcal{M},~{}\mathcal{N}\in{{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ and let $\eta\in Hom_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{N})$. Then, there is a morphism $\rho(\eta)\in Hom_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{N}\otimes H)$ determined by setting

[TABLE]

for any $X\in Ob(\mathcal{D})$ and $m\in\mathcal{M}(X)$.

Proof.

Using (5.1) and the fact that $\eta\in Hom_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{N})$, we have

[TABLE]

for any $f\in Hom_{\mathcal{D}}(X,Y)$ and $m\in\mathcal{M}(X)$. ∎

We now recall the notion of a rational left $H^{*}$-module (see, for instance, [SCS]) which will be used in the next result. Given a left $H^{*}$-module $M$, there is a morphism $\rho_{M}:M\longrightarrow Hom_{K}(H^{*},M)$ corresponding to the canonical morphism $H^{*}\otimes M\longrightarrow M$. There is an obvious inclusion $M\otimes H\hookrightarrow Hom_{K}(H^{*},M)$ given by $(m\otimes h)(h^{*})=h^{*}(h)m$ for any $m\in M$, $h\in H$ and $h^{*}\in H^{*}$.

Definition 5.12**.**

(see [SCS, Definition 2.2.2]) A left $H^{*}$-module $M$ is said to be rational if $\rho_{M}(M)\subseteq M\otimes H$, where $M\otimes H$ is viewed as a subspace of $Hom_{K}(H^{*},M)$. The full subcategory of rational $H^{*}$-modules will be denoted by $Rat(H^{*}\text{-}Mod)$ .

If $M$ is a right $H$-comodule with $H$-coaction $m\mapsto\sum m_{0}\otimes m_{1}$, then $M$ becomes a left $H^{*}$-module via the action $h^{*}m:=\sum h^{*}(m_{1})m_{0}$ for $h^{*}\in H^{*}$ and $m\in M$. This determines a functor

[TABLE]

It is well known (see [SCS, Theorem 2.2.5]) that this functor defines an equivalence of categories between $Comod\text{-}H$ and the subcategory $Rat(H^{*}\text{-}Mod)$ of $H^{*}\text{-}Mod$.

Proposition 5.13**.**

Let $\mathcal{M},~{}\mathcal{N}\in{{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ and suppose that $\mathcal{M}$ is finitely generated as an object in $\mathcal{D}\text{-}Mod$. Then, $Hom_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{N})$ is a right $H$-comodule. In particular, $HOM_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{N})=Hom_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{N})$.

Proof.

Since $\mathcal{M}$ is finitely generated in $\mathcal{D}\text{-}Mod$, by Proposition 5.5, there exists a finite dimensional $H$-comodule $W$ and an epimorphism

[TABLE]

in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$, for finitely many objects $\{X_{i}\}_{i\in I}$ in $\mathcal{D}$. From the description of epimorphisms in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ in (5.2), we know that $\eta$ is also an epimorphism in $\mathcal{D}\text{-}Mod$. The map

[TABLE]

is therefore a monomorphism for each $\mathcal{N}\in{{}_{\mathcal{D}}}{\mathscr{M}}^{H}$. Using the fact that $\eta(Y)$ is $H$-colinear for each $Y\in Ob(\mathcal{D})$, we will now verify that the morphism $Hom(\eta,\mathcal{N})$ is $H^{*}$-linear. For any $h^{*}\in H^{*}$, $\xi\in Hom_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{N})$, $Y\in Ob(\mathcal{D})$ and $\tilde{f}\otimes w\in\left(\bigoplus_{i\in I}{{}_{X_{i}}}{\bf h}(Y)\right)\otimes W$, we have

[TABLE]

This shows that $Hom_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{N})$ is an $H^{*}$-submodule of $\bigoplus_{i\in I}Hom_{\mathcal{D}\text{-}Mod}({{}_{X_{i}}}{\bf h}\otimes W,~{}\mathcal{N})$.

For each $i\in I$, we now prove that $\rho:Hom_{\mathcal{D}\text{-}Mod}({{}_{X_{i}}}{\bf h}\otimes W,\mathcal{N})\longrightarrow Hom_{\mathcal{D}\text{-}Mod}({{}_{X_{i}}}{\bf h}\otimes W,\mathcal{N}\otimes H)$, as defined in (5.6), gives an $H$-comodule structure on $Hom_{\mathcal{D}\text{-}Mod}({{}_{X_{i}}}{\bf h}\otimes W,\mathcal{N})$. Since $W$ is finite dimensional, we have

[TABLE]

This gives a well defined morphism

[TABLE]

We will verify that (5.7) gives a right $H$-coaction. For this, we need to show that for any $\zeta\in Hom_{\mathcal{D}\text{-}Mod}({{}_{X_{i}}}{\bf h}\otimes W,~{}\mathcal{N})$, we have $(\rho\otimes\text{id})\rho(\zeta)=(\text{id}\otimes\Delta)\rho(\zeta)$ and $(\text{id}\otimes\varepsilon)\rho(\zeta)=\zeta$. The latter equality is easy to verify. By (5.7), we know that $\rho(\zeta)=\sum\zeta_{0}\otimes\zeta_{1}\in Hom_{\mathcal{D}\text{-}Mod}({{}_{X_{i}}}{\bf h}\otimes W,~{}\mathcal{N})\otimes H$. Thus, for any $X\in Ob(\mathcal{D})$ and $u\in{{}_{X_{i}}}{\bf h}(X)\otimes W$, we have

[TABLE]

The third equality above follows by applying $\rho_{\mathcal{N}(X)}\otimes id_{H}$ on the equality $\sum\zeta_{0}(X)(u_{0})\otimes\zeta_{1}=\rho(\zeta)(X)(u_{0})$ and the last one is obtained by applying $id_{H}\otimes\Delta$ on \sum\zeta_{0}(X)(u)\otimes\zeta_{1}=\sum\big{(}\zeta(X)(u_{0})\big{)}_{0}\otimes S^{-1}(u_{1})\big{(}\zeta(X)(u_{0})\big{)}_{1}. Thus, we have shown that $Hom_{\mathcal{D}\text{-}Mod}({{}_{X_{i}}}{\bf h}\otimes W,~{}\mathcal{N})$ is a right $H$-comodule.

Moreover, the $H^{*}$-action on $Hom_{\mathcal{D}\text{-}Mod}({{}_{X_{i}}}{\bf h}\otimes W,~{}\mathcal{N})$ as in (5.5) is given precisely by the $H$-coaction as in (5.6). Therefore, $Hom_{\mathcal{D}\text{-}Mod}({{}_{X_{i}}}{\bf h}\otimes W,~{}\mathcal{N})$ is a rational $H^{*}$-module. Since the category of rational $H^{*}$-modules contains direct sums (it is equivalent to $Comod\text{-}H$), it follows that $\bigoplus_{i\in I}Hom_{\mathcal{D}\text{-}Mod}({{}_{X_{i}}}{\bf h}\otimes W,~{}\mathcal{N})$ is also a rational $H^{*}$-module. Being an $H^{*}$-submodule of $\bigoplus_{i\in I}Hom_{\mathcal{D}\text{-}Mod}({{}_{X_{i}}}{\bf h}\otimes W,~{}\mathcal{N})$, it is now clear that $Hom_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{N})$ is also a rational left $H^{*}$-module and hence a right $H$-comodule.

It may be verified that the functor $Hom_{\mathcal{D}\text{-}{Mod}}(\mathcal{M},{-}):{{}_{\mathcal{D}}}{\mathscr{M}}^{H}\longrightarrow Comod$-$H$ is right adjoint to the functor $\mathcal{M}\otimes{-}:Comod$-$H\longrightarrow{{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ given by $N\mapsto\mathcal{M}\otimes N$. Thus, by the uniqueness of adjoints, we have $Hom_{\mathcal{D}\text{-}{Mod}}(\mathcal{M},{-})=HOM_{\mathcal{D}\text{-}{Mod}}(\mathcal{M},{-})$. ∎

A morphism $\mathcal{N}\longrightarrow\mathcal{N^{\prime}}$ in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ induces a morphism of functors $\mathcal{N}\otimes{(-)}\longrightarrow\mathcal{N^{\prime}}\otimes{(-)}$ and hence a morphism $\mathscr{R}_{\mathcal{N^{\prime}}}\longrightarrow\mathscr{R}_{\mathcal{N}}$ of their respective right adjoints. Thus, for any $\mathcal{L}\in{{}_{\mathcal{D}}}{\mathscr{M}}^{H}$, we have a functor $HOM_{\mathcal{D}\text{-}Mod}({-},\mathcal{L}):({{}_{\mathcal{D}}}{\mathscr{M}}^{H})^{op}\longrightarrow Comod$-$H$ which takes $\mathcal{N}$ to $HOM_{\mathcal{D}\text{-}Mod}(\mathcal{N},\mathcal{L})=\mathscr{R}_{\mathcal{N}}(\mathcal{L})$.

Proposition 5.14**.**

(1) For any $\mathcal{L}\in{{}_{\mathcal{D}}}{\mathscr{M}}^{H}$, the functor $HOM_{\mathcal{D}\text{-}Mod}({-},\mathcal{L}):({{}_{\mathcal{D}}}{\mathscr{M}}^{H})^{op}\longrightarrow Comod$-$H$ is left exact, i.e., it preserves kernels.

(2) If $\mathcal{I}$ is injective in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$, then $HOM_{\mathcal{D}\text{-}Mod}({-},\mathcal{I})$ is exact.

(3) If $\mathcal{I}$ is injective in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$, then $HOM_{\mathcal{D}\text{-}Mod}({-},\mathcal{I})$ takes every short exact sequence in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ to a split short exact sequence in $Comod\text{-}H$.

Proof.

(1) Let $\eta:\mathcal{M}\longrightarrow\mathcal{N}$ be a morphism in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ and let $\mathcal{P}:=Coker(\eta)$. Then, for any $T\in Comod\text{-}H$, $Coker(\eta\otimes{id}_{T}:\mathcal{M}\otimes T\longrightarrow\mathcal{N}\otimes T)=\mathcal{P}\otimes T$. From the adjunction in (5.4), we now have

[TABLE]

for any $T\in Comod$-$H$. From Yoneda Lemma, it follows that

[TABLE]

(2) Let $0\longrightarrow\mathcal{M}\longrightarrow\mathcal{N}\longrightarrow\mathcal{P}\longrightarrow 0$ be a short exact sequence in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$. From (1), we already know that

[TABLE]

is exact. We need to show that $q$ is an epimorphism. For any $T\in Comod\text{-}H$, we notice that $0\longrightarrow\mathcal{M}\otimes T\longrightarrow\mathcal{N}\otimes T\longrightarrow\mathcal{P}\otimes T\longrightarrow 0$ is still a short exact sequence in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$. If $\mathcal{I}$ is an injective object in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$, we see that

[TABLE]

is an exact sequence of $K$-vector spaces. Using the adjunction in (5.4), it follows that

[TABLE]

is short exact in $Vect_{K}$. By setting $T=HOM_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{I})$ in (5.9), we see that there exists a morphism $f:HOM_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{I})\longrightarrow HOM_{\mathcal{D}\text{-}Mod}(\mathcal{N},\mathcal{I})$ of $H$-comodules such that $q\circ f$ is the identity on $HOM_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{I})$. This shows that $q:HOM_{\mathcal{D}\text{-}Mod}(\mathcal{N},\mathcal{I})\longrightarrow HOM_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{I})$ is an epimorphism. The result of (3) is clear from the proof of (2). ∎

Proposition 5.15**.**

Let $\mathcal{D}$ be a left noetherian right co-$H$-category and let $\mathcal{M}\in{{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ be finitely generated as an object in $\mathcal{D}$-$Mod$. If $\mathcal{I}$ is an injective object in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$, then $\textnormal{Ext}_{\mathcal{D}\text{-}Mod}^{p}(\mathcal{M},\mathcal{I})=0$ for all $p>0$.

Proof.

Since $\mathcal{M}\in{{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ is finitely generated in $\mathcal{D}$-$Mod$, by Proposition 5.5, there exists a finite dimensional $H$-comodule $W_{0}$ and an epimorphism

[TABLE]

in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ for finitely many objects $\{X_{i}\}_{1\leq i\leq n_{0}}$ in $\mathcal{D}$. Then, $\mathcal{K}:=Ker(\eta_{0})$ is a subobject of $\mathcal{P}_{0}$ in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$. Since $\mathcal{D}$ is left noetherian, $\mathcal{P}_{0}$ is a noetherian left $\mathcal{D}$-module (see, for instance, [Mit1, $\S$ 3]). Thus, the submodule $\mathcal{K}=Ker(\eta_{0})$ of $\mathcal{P}_{0}$ is finitely generated as an object in $\mathcal{D}$-$Mod$. Therefore, we obtain a finite dimensional $H$-comodule $W_{1}$ and an epimorphism

[TABLE]

in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ for finitely many objects $\{Y_{j}\}_{1\leq j\leq n_{1}}$ in $\mathcal{D}$. Since $W_{0}$ and $W_{1}$ are finite dimensional $K$-vector spaces, clearly $\mathcal{P}_{0}$ and $\mathcal{P}_{1}$ are free left $\mathcal{D}$-modules. Moreover, $Im(\eta_{1})=\mathcal{K}=Ker(\eta_{0})$. Continuing in this way, we can construct a free resolution of the module $\mathcal{M}$ in the category $\mathcal{D}$-$Mod$:

[TABLE]

This gives us

[TABLE]

Since $\mathcal{M}$ and $\{\mathcal{P}_{i}\}_{i\geq 0}$ are finitely generated in $\mathcal{D}$-$Mod$, it follows from Proposition 5.13 that $HOM_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{I})=Hom_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{I})$ and $HOM_{\mathcal{D}\text{-}Mod}(\mathcal{P}_{i},\mathcal{I})=Hom_{\mathcal{D}\text{-}Mod}(\mathcal{P}_{i},\mathcal{I})$. From Proposition 5.14, we know that the functor $HOM_{\mathcal{D}\text{-}Mod}(-,\mathcal{I})$ is exact and it follows that $\textnormal{Ext}_{\mathcal{D}\text{-}Mod}^{p}(\mathcal{M},\mathcal{I})=H^{p}(HOM_{\mathcal{D}{\text{-}}Mod}(\mathcal{P}_{*},\mathcal{I}))=0$ for all $p>0$. ∎

Proposition 5.16**.**

Let $\mathcal{D}$ be a left noetherian right co-$H$-category. Let $\mathcal{M},\mathcal{N}\in{{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ with $\mathcal{M}$ finitely generated as an object in $\mathcal{D}$-$Mod$. If $\mathcal{E}^{*}$ is an injective resolution of $\mathcal{N}$ in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$, then

[TABLE]

Proof.

Let $\mathcal{P}_{*}$ be the free resolution of $\mathcal{M}$ in $\mathcal{D}$-$Mod$ constructed as in the proof of Proposition 5.15. Then, we have

[TABLE]

where the second equality follows from Proposition 5.13. Since $HOM_{\mathcal{D}{\text{-}}Mod}(\mathcal{P}_{*},\mathcal{N})$ is a complex in $Comod$-$H$ and $Comod$-$H$ is an abelian category, it follows that H^{p}\big{(}HOM_{\mathcal{D}{\text{-}}Mod}(\mathcal{P}_{*},\mathcal{N})\big{)}\in Comod-$H$. Hence, we may consider the family $\{\textnormal{Ext}_{\mathcal{D}\text{-}Mod}^{p}(\mathcal{M},-)\}_{p\geq 0}$ as a $\delta-$functor from ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ to $Comod$-$H$.

By Proposition 5.15, $\textnormal{Ext}_{\mathcal{D}\text{-}Mod}^{p}(\mathcal{M},\mathcal{I})=0$, $p>0$ for every injective object $\mathcal{I}\in{{}_{\mathcal{D}}}{\mathscr{M}}^{H}$. Since ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ has enough injectives, it follows that each $\textnormal{Ext}_{\mathcal{D}\text{-}Mod}^{p}(\mathcal{M},-):{{}_{\mathcal{D}}}{\mathscr{M}}^{H}\longrightarrow Comod$-$H$ is effaceable (see, for instance, [Hart, $\S$ III.1]).

Since ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ has enough injectives, we can consider the right derived functors

[TABLE]

For $p=0$, we notice that $\textnormal{Ext}_{\mathcal{D}\text{-}Mod}^{0}(\mathcal{M},-)=Hom_{\mathcal{D}\text{-}Mod}(\mathcal{M},-)=HOM_{\mathcal{D}{\text{-}}Mod}(\mathcal{M},-)=R^{0}HOM_{\mathcal{D}{\text{-}}Mod}(\mathcal{M},-)$ as functors from ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ to $Comod\text{-}H$. Since each $\textnormal{Ext}_{\mathcal{D}\text{-}Mod}^{p}(\mathcal{M},-)$ is effaceable for $p>0$, we see that the family $\{\textnormal{Ext}_{\mathcal{D}\text{-}Mod}^{p}(\mathcal{M},-)\}_{p\geq 0}$ forms a universal $\delta$-functor and it follows from [Hart, Corollary III.1.4] that

[TABLE]

for every $p\geq 0$. Therefore, we have

[TABLE]

∎

Recall that by Proposition 5.13, for any $\mathcal{M}\in{{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ with $\mathcal{M}$ finitely generated as an object in $\mathcal{D}$-$Mod$, we have $Hom_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{N})=HOM_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{N})\in Comod$-$H$.

Theorem 5.17**.**

Let $\mathcal{D}$ be a left noetherian right co-$H$-category. Let $\mathcal{M}\in{{}_{\mathcal{D}}}{\mathscr{M}}^{H}$ with $\mathcal{M}$ finitely generated as an object in $\mathcal{D}$-$Mod$. We consider the functors

[TABLE]

Then, we have the following spectral sequence

[TABLE]

Proof.

By Corollary 5.7, we have

[TABLE]

By definition,

[TABLE]

where $\{\mathcal{E}^{*}\}$ is an injective resolution of $\mathcal{N}$ in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$. Applying Corollary 5.16, we obtain $\textnormal{Ext}_{\mathcal{D}\text{-}Mod}^{q}(\mathcal{M},\mathcal{N})=R^{q}\mathscr{F}(N)$. For any injective object $\mathcal{I}$ in ${{}_{\mathcal{D}}}{\mathscr{M}}^{H}$, we know that $\mathscr{F}(\mathcal{I})=Hom_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{I})=HOM_{\mathcal{D}\text{-}Mod}(\mathcal{M},\mathcal{I})$ is injective in $Comod\text{-}H$ by Corollary 5.8. Since $Comod$-$H$ is a Grothendieck category, it has enough injectives. The result now follows from Grothendieck spectral sequence for composite functors (see [Grothen]). ∎