On the cohomology of locally finite $A\#_HC$-modules

Abstract

Let $k$ be a field, $H$ a Hopf algebra with bijective antipode, $A$ a left $H$-module algebra and $C$ a left $H$-comodule algebra. The categories of $C$-locally finite $C$-modules and $C$-locally finite $A\#_HC$-modules are abelian with enough injectives. Assuming that there are algebra morphisms $\phi : H \rightarrow C$ and $\epsilon _C : C \rightarrow k$ subjected to some conditions, we study the right derived functors of the associated Hom functors, and of the $C$-invariants functor. Then we derive spectral sequences that connect them. We also discuss when the $C$-invariants functor preserves injectives. In an appendix we establish some of these results in the category of $C$-locally finite Doi–Hopf modules.

Appendix: Locally finite Doi–Hopf modules: the derived functors of \({\mathcal L}^D(- , -)\) and \({}_{C}Hom^D(- , -)\)

In this section, $H$ is a Hopf algebra with bijective antipode, $C$ is a left $H$-comodule algebra. We keep the conventions and notations of the preceding sections. Let $D$ be a coalgebra. The linear dual $D^*$ of $D$ is an associative algebra with identity under the convolution product. A left $D^*$ module $M$ is called rational if there exist finite elements $d_1, d_2, \ldots , d_l \in D$ and $m_1, m_2, \ldots , m_l \in M$ such that $d^*m=\sum <d^* , d_i>m_i$. A coalgebra $D$ is a right $H$-module coalgebra if $D$ is a right $H$-module, and $\Delta _D$ and $\epsilon _D$ are $H$-linear; i.e.,

$$\begin{aligned}\Delta _D(d\cdot h)=d_1\cdot h_1 \otimes d_2\cdot h_2 \quad \hbox {and} \quad \epsilon _D(d\cdot h)=\epsilon _H(h)\epsilon _D(d).\end{aligned}$$

A right $D$-comodule is a vector space $M$ with a $k$-linear map ${\rho }_{M , D} : M \rightarrow M \otimes D$ such that

$$\begin{aligned} \rho _{M , D}(m)=m_{0} \otimes m_{1} \in M\otimes D \quad \text{ and } \quad \epsilon _D(m_{1})m_{0} =m.\end{aligned}$$

A $k$-linear map $f: M \rightarrow N$ between two right $D$-comodules is called a morphism of $D$-comodules or a $D$-colinear map if

$$\begin{aligned}{\rho }_{N , D} \circ f =(f \otimes id_D) \circ {\rho }_{M , D}\end{aligned}$$

or equivalently,

$$\begin{aligned}{f(m)}_{0} \otimes {f(m)}_{1} = f(m_{0} ) \otimes m_{1} \quad \hbox {for all} \quad m \in M.\end{aligned}$$

We will denote by $\mathcal{M}^D$ the category of right $D$-comodules. The morphisms of $\mathcal{M}^D$ are the $D$-colinear maps. The vector space of $D$-colinear maps between two right $D$-comodules $M$ and $N$ will be denoted $Hom^D(M, N)$.

Let $D$ be a right $H$-module coalgebra. We call the three-tuples $(H , C , D)$ a left-right Doi–Hopf datum. The corresponding category ${_C{\mathcal{M}(H)}^D}$ of left-right $(H , C , D)$-Doi–Hopf modules (see Caenepeel et al. 1997) is the category whose objects are left $C$-modules and right $D$-comodules satisfying the compatibility condition

$$\begin{aligned} \rho _{M , D}(cm)=c_{(0)}m_{0} \otimes m_{1}.S_H(c_{(-1)}). \end{aligned}$$

The morphisms of ${_C{\mathcal{M}(H)}^D}$ are the $C$-linear and the $D$-colinear maps. For $M$ and $N$ in ${_C{\mathcal{M}(H)}^D}$, we have

$$\begin{aligned} {}_CHom^D(M , N)=Hom^D(M , N) \cap {_C}Hom(M , N). \end{aligned}$$

By Brzezinski and Wisbauer (2003) or Caenepeel et al. (1997), ${_C{\mathcal{M}(H)}^D}$ is a Grothendieck category with enough injectives.

For every $M$ in $\mathcal{M}^D$, $C \otimes M$ is an object of ${_C{\mathcal{M}(H)}^D}$: the $C$-action is the obvious one while the $D$-coaction is given by $\rho _{M , D}(c \otimes m)=c_{(0)} \otimes m_{0} \otimes m_{1}.S_H(c_{(-1)})$.

Lemma 4.25

Let $M$ be a right $D$-comodule and $N$ an $(H,C, D)$-Doi–Hopf module. The natural $k$-isomorphism $\psi : {_C}Hom(C \otimes M, N) \rightarrow Hom(M,N)$ defined by $\psi (f)(m)=f(1 \otimes m)$ induces a $k$-isomorphism $\psi : {_C}Hom^D(C \otimes M, N) \rightarrow Hom^D(M , N)$.

Proof

If $f \in {_C}Hom^D(C \otimes M, N)$, then

$$\begin{aligned}\begin{array}{rcl}\psi (f)(m)_0 \otimes \psi (f)(m)_1&{}=&{}f(1 \otimes m)_0 \otimes f(1 \otimes m)_1\\ &{}=&{}f((1 \otimes m)_0) \otimes (1 \otimes m)_1\\ &{}=&{}f(1 \otimes m_0) \otimes m_1\\ &{}=&{}\psi (f)(m_0) \otimes m_1.\end{array}\end{aligned}$$

So $\psi (f)$ is $D$-colinear. If $g \in Hom^D(M , N)$, then

$$\begin{aligned}\begin{array}{rcl}\psi ^{-1}(g)(c \otimes m)_0 \otimes \psi ^{-1}(g)(c \otimes m)_1=(cg(m))_0 \otimes (cg(m))_1=\\ c_{(0)}g(m_0) \otimes m_1.S_H(c_{(-1)})=\psi ^{-1}(g)((c \otimes m)_0) \otimes (c \otimes m)_1.\end{array}\end{aligned}$$

So $\psi ^{-1}(g)$ is $D$-colinear. $\square$

Corollary 4.26

Every injective in ${_C{\mathcal{M}(H)}^D}$ is injective in $\mathcal{M}^D$.

By Caenepeel et al. (1997), $D^*$ is a left $H$-module algebra: the action is given by

$$\begin{aligned}<h\cdot d^* , d'>=<d^* , d'\cdot h>, \forall h \in H, d' \in D.\end{aligned}$$

Every right $D$-comodule $M$ is a rational left $D^*$-module: the action is given by $d^*\cdot m=<d^* , m_1>m_0$, and $\mathcal{M}^D$ is a full subcategory of ${}_{D^*}\mathcal{M}$; that is,

$$\begin{aligned}Hom^D(M , N)={_{D^*}}Hom(M , N), \forall M, N \quad \in \mathcal{M}^D.\end{aligned}$$

Let us consider the smash product $D^*\#_HC$. By Caenepeel et al. (1997), every left-right $(H, C , D)$-Doi–Hopf module is a rational left $D^*\#_HC$-module and ${_C{\mathcal{M}(H)}^D}$ is a full subcategory of ${}_{D^* \#_HC}\mathcal{M}$, that is,

$$\begin{aligned}{}_CHom^D(M , N)={_{D^* \#_{H}C}}Hom(M , N), \forall \quad M, N \in {_C}{\mathcal{M}(H)}^D.\end{aligned}$$

It follows that an object of ${_C{\mathcal{M}(H)}^D}$ that is injective in ${}_{D^* \#_HC}\mathcal{M}$ is injective in ${_C{\mathcal{M}(H)}^D}$. Note that an injective of ${_C{\mathcal{M}(H)}^D}$ is not necessarily an injective of ${}_{D^* \#_HC}\mathcal{M}$.

By Caenepeel et al. (1997), every rational left $D^*\#_HC$-module is an object of ${_C{\mathcal{M}(H)}^D}$.

From now on, we assume that there is an algebra morphism $\phi : H \rightarrow C$. Let us recall that if $\phi$ is $H$-colinear, then the conditions $(\beta _1)$ and $(\beta _2)$ are satisfied.

The following lemma is a consequence of Lemma 3.7.

Lemma 4.27

Assume that condition $(\beta _2)$ is satisfied. Let $M$ and $N$ be objects of ${_C{\mathcal{M}(H)}^D}$. Then $Hom^D(M , N)$ is a $C$-submodule of $Hom(M , N)$. If we assume that there is an algebra morphism $\epsilon _C : C \rightarrow k$, then ${}_{C}Hom^D(M, N)=Hom^D(M , N)^C$.

For $M$ and $N$ in ${_C}{\mathcal{M}(H)}^D$, we set ${\mathcal L}^D(M, N)=Hom^D(M, N) \cap {_{(C)}}Hom(M, N)$.

If condition $(\beta _2)$ is satisfied, Lemma 4.27 implies that ${\mathcal L}^D(M, N)$ is a $C$-locally finite $C$-submodule of $Hom^D(M, N)$.

Lemma 4.28

Let $M$ be in ${_C}\mathcal{M}$ and $N$ in ${_C{\mathcal{M}(H)}^D}$. Assume that condition $(\beta _1)$ is satisfied. Then $N\otimes M$ is an object of ${_C{\mathcal{M}(H)}^D}$: the coaction is given by $\rho _{M , D}(n \otimes m)=n_{0} \otimes m \otimes n_{1}$.

Proof

$N$ is a rational left $D^*$-module. By Lemma 3.10(1), $N\otimes M$ is a left $D^*\#_H$ $C$-module. It is easy to show that the $D^*$-action is rational. $\square$

Lemma 4.29

Assume that conditions $(\beta _1)$ and $(\beta _2)$ are satisfied. Let $N$, $P$ be $(H , C , D)$-Doi–Hopf modules and $M$ a left $C$-module.

• (1) The $k$-isomorphism $\varphi : {_{C}}Hom(N\otimes M , P)\rightarrow {_{C}}Hom(M, Hom(N, P))$ defined by $\varphi (f)(m)(n)=f(n\otimes m)$ (see Lemma 2.5) induces a $k$-isomorphism

  $$\begin{aligned}\varphi : {_C}Hom^D(N\otimes M, P) \rightarrow {_C}Hom(M , Hom^D(N , P)).\end{aligned}$$

• (2) If $M$ is $C$-locally finite, $\varphi$ induces a $k$-isomorphism

  $$\begin{aligned}{}_{C}Hom^D(N \otimes M, P) \rightarrow {{_C}}Hom(M, {\mathcal L}^D(N , P)).\end{aligned}$$

Proof

(1) We know that ${}_CHom^D(N \otimes M , P)={_{D^* \#_HC}}Hom(N \otimes M , P)$ and $Hom^D(N , P)={_{D^*}}Hom(N , P)$. The result follows from Lemma 3.11.

(2) Follows from (1). $\square$

Corollary 4.30

Assume that conditions $(\beta _1)$ and $(\beta _2)$ are satisfied. Let $N$ be an object of ${}_C{\mathcal{M }(H)}^D$.

• (1) If $I$ is an injective in ${_C{\mathcal{M}(H)}^D}$, then $Hom^D(N, I)$ is an injective in ${}_{C}\mathcal{M}$.

• (2) If $I$ is an injective in ${_C{\mathcal{M}(H)}^D}$, then ${\mathcal L}^D(N, I)$ is an injective in ${}_{(C)}\mathcal{M}$.

Proof

• (1) Follows from Lemma 4.29(1) and from the exactness of the functor $N\otimes (-)$ defined from ${}_C\mathcal{M}$ to ${}_C{\mathcal{M}(H)}^D$.

• (2) Follows from Lemma 4.29(2) and from the exactness of the functor $N\otimes (-)$ defined from ${}_{(C)}\mathcal{M}$ to ${}_C{\mathcal{M}(H)}^D$. $\square$

Let us denote by:

• ${{\mathcal L}^D}^p(- , -)$ the right derived functors of ${\mathcal L}^D(- , -)$ defined on ${_C{\mathcal{M}(H)}^D} \times {_C{\mathcal{M}(H)}^D}$

• ${Ext^D}^p(- , -)$ the right derived functors of $Hom^D(-, -)$ defined on $\mathcal{M}^D \times \mathcal{M}^D$

• ${}_{C}{Ext^D}^p(- , -)$ the right derived functors of ${}_CHom^D(-, -)$ defined on ${_C{\mathcal{M}(H)}^D} \times {_C{\mathcal{M}(H)}^D}$.

Proposition 4.31

Assume that conditions $(\beta _1)$ and $(\beta _2)$ are satisfied. Let $N$ and $P$ be objects of ${}_C{\mathcal{M}(H)}^D$ and $M$ an object of ${}_C\mathcal{M}$. Then we have the spectral sequence

$$\begin{aligned}{}_CExt^p(M , {Ext^D}^q(N , P)) \Rightarrow {_C}{Ext^D}^{p+q}(N \otimes M, P).\end{aligned}$$

Proof

By Corollary 4.26 every injective in ${}_C{\mathcal{M}(H)}^D$ is injective in $\mathcal{M}^D$. By Lemma 4.29(1), the functor ${}_{C}Hom^D(N \otimes M,-)$ defined from ${}_C{\mathcal{M}(H)}^D$ to $\mathcal{M}$ is the composition of the functors $Hom^D(N , -)$ defined from ${}_C{\mathcal{M}(H)}^D$ to ${}_C\mathcal{M}$ and ${}_CHom(M , -)$ defined from ${}_C\mathcal{M}$ to $\mathcal{M}$. By Corollary 4.30(1), the functor $Hom^D(N , -)$ carries injectives from ${}_C{\mathcal{M}(H)}^D$ to injectives of ${}_C\mathcal{M}$. The result follows from the Grothendieck spectral sequence for composite functors. $\square$

Taking $M=k$ in Proposition 4.31, we get

Corollary 4.32

Assume that conditions $(\beta _1)$ and $(\beta _2)$ are satisfied and that there is an algebra morphism $\epsilon _C : C \rightarrow k$. Let $N$ and $P$ be objects of ${}_C{\mathcal{M}(H)}^D$. Then we have the spectral sequences

$$\begin{aligned}{}_CH^p({Ext^D}^q(N , P)) \Rightarrow {_C}{Ext^D}^{p+q}(N , P).\end{aligned}$$

Proposition 4.33

Assume that conditions $(\beta _1)$ and $(\beta _2)$ are satisfied. Let $N$ and $P$ be objects of ${_C{\mathcal{M}(H)}^D}$ and $M$ an object of ${}_{(C)}\mathcal{M}$. Then we have the spectral sequence

$$\begin{aligned}{}_C{\widetilde{Ext}}^p(M , {\mathcal{L}^D}^q(N , P)) \Rightarrow {_C}{Ext^D}^{p+q}(N \otimes M, P).\end{aligned}$$

Proof

By Lemma 4.29(2), the functor ${}_{C}Hom^D(N \otimes M, -)$ defined from ${}_C{\mathcal{M}(H)}^D$ to $\mathcal{M}$ is the composition of the functors $\mathcal{L}^D(N , -)$ defined from ${}_C{\mathcal{M}(H)}^D$ to ${}_{(C)}\mathcal{M}$ and ${}_CHom(M , -)$ defined from ${}_{(C)}\mathcal{M}$ to $\mathcal{M}$. By Corollary 4.30(2), the functor $\mathcal{L}^D(N , -)$ carries injectives from ${}_C{\mathcal{M}(H)}^D$ to injectives of ${}_{(C)}\mathcal{M}$. The result follows from the Grothendieck spectral sequence for composite functors. $\square$

Taking $M=k$ in Proposition 4.33, we get the following result.

Corollary 4.34

Assume that conditions $(\beta _1)$ and $(\beta _2)$ are satisfied and that there is an algebra morphism $\epsilon _C : C \rightarrow k$. Let $N$ and $P$ be objects of ${}_C{\mathcal{M}(H)}^D$. Then we have

$$\begin{aligned}R^pa(C , {\mathcal{L}^D}^q(N , P)) \Rightarrow {_C}{Ext^D}^{p+q}(N , P)\end{aligned}$$

Corollary 4.35

Assume that conditions $(\beta _1)$ and $(\beta _2)$ are satisfied and that $k$ is projective in ${}_C\mathcal{M}^{fd}$. Let $N$ and $P$ be objects of ${}_C{\mathcal{M}(H)}^D$ and $M$ an object of ${}_{(C)}\mathcal{M}$. Then we have

$$\begin{aligned}{}_CHom(M , {\mathcal{L}^D}^q(N , P))={_C}{Ext^D}^q(N \otimes M, P).\end{aligned}$$

Assume that there is an algebra morphism $\epsilon _C : C \rightarrow k$. Then we have

$$\begin{aligned}{\mathcal{L}^D}^q(N , P)^C={_{C}}{Ext^D}^q(N , P)\end{aligned}$$

Proof

${}_{(C)}\mathcal{M}$ is a semisimple category. The spectral sequence of Proposition 4.33 collapses, and we get the first assertion. Taking $M=k$ in the first assertion,we get the second one. $\square$

Proposition 4.36

Assume that conditions $(\beta _1)$ and $(\beta _2)$ are satisfied. Let $N$ and $P$ be objects of ${_C{\mathcal{M}(H)}^D}$. Then we have the spectral sequence

$$\begin{aligned}{}_{(C)}H^p({Ext^D}^q(N , P)) \Rightarrow {{\mathcal L}^D}^{p+q}(N , P).\end{aligned}$$

Proof

By Corollary 4.26, every injective in ${}_C{\mathcal{M}(H)}^D$ is injective in $\mathcal{M}^D$. We have $\mathcal{L}^D(N , -)=Hom^D(N , -)^{(C)}$ on ${}_C{\mathcal{M}(H)}^D$. By Corollary 4.30(1), the functor $Hom^D(N , -)$ carries injectives from ${_C{\mathcal{M}(H)}^D}$ to injectives of ${}_{C}\mathcal{M}$. $\square$

Remark 4.37

• (1) If $C=H$ is considered as a left $H$-comodule algebra via $\Delta _H$, then ${}_H{\mathcal{M}(H)}^D$ is the category ${}_H\mathcal{M}^D$ of anti-$[D , H]$-Hopf modules, i.e.; the left $H$-modules and right $D$-comodules $M$ such that $(hm)_0 \otimes (hm)_1=h_2m_0 \otimes m_1S_H(h_1)$ for all $m \in M$. In this situation, we can take $\epsilon _C = \epsilon _H$. We know by Lemma 2.15 that conditions $(\beta _1)$, $(\beta _2)$ are satisfied with $\phi =id_H$.

• (2) Let us consider $H$ as a left $H^{op} \otimes H$-comodule algebra via

  $$\begin{aligned}\rho _{H^{op} \otimes H , H}(h)=h_{(-1)} \otimes h_{(0)}=S_H(h_3) \otimes h_1\otimes h_2\end{aligned}$$

  and as a right $H^{op} \otimes H$-module coalgebra via $l \leftharpoonup (h \otimes h')=hlh'$. Then ${}_H{\mathcal{M}(H^{op} \otimes H)}^H$ is the category of left-right anti-Yetter-Drinfeld-modules; i.e., the left $H$-modules and right $H$-comodules $M$ such that $(hm)_0 \otimes (hm)_1=h_2m_0 \otimes h_3m_1S_H(h_1)$ for all $m \in M$. In this situation, we can take $\epsilon _C = \epsilon _H$. If $H$ is cocommutative, we know by Lemma 2.15 that conditions $(\beta _1)$, $(\beta _2)$ are satisfied with $\phi =\epsilon _H \otimes id_H$.