Cohomology of modules over H -categories and co- H -categories
aa r X i v : . [ m a t h . R A ] J u l Cohomology of modules over H -categories and co- H -categories Mamta Balodi ∗† Abhishek Banerjee ‡§ Samarpita Ray ¶ Department of Mathematics, Indian Institute of Science, Bangalore 560012, India.
Abstract
Let H be a Hopf algebra. We consider H -equivariant modules over a Hopf module category C as modulesover the smash extension C H . We construct Grothendieck spectral sequences for the cohomologies as wellas the H -locally finite cohomologies of these objects. We also introduce relative (D , H ) -Hopf modules over aHopf comodule category 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 objectsand higher derived functors of coinvariants.
MSC(2010) Subject Classification: 16S40, 16T05, 18E05Keywords: H -categories, co- H -categories, H -equivariant modules, relative Hopf modules Let H be a Hopf algebra over a field K . An H -category is a small K -linear category C such that the morphismspace Hom C ( X, Y ) is an H -module for each couple of objects X , Y ∈ Ob (C) and the composition of morphismsin C is well-behaved with respect to the action of H . Similarly, a co- H -category is a small K -linear category D such that the morphism space Hom D ( X, Y ) is an H -comodule for each couple of objects X , Y ∈ Ob (D) and the composition of morphisms in 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 themonoidal category of H -comodules. The purpose of this paper is to study cohomology in module categoriesover H -categories and co- H -categories.The Hopf module categories that we use were first considered by Cibils and Solotar [6], where they discovereda Morita equivalence that relates Galois coverings of a category to its smash extensions via a Hopf algebra. Weview these H -categories and the modules over them as objects of interest in their own right. We recall here thatan ordinary ring may be expressed as a preadditive category with a single object. Accordingly, an arbitrarysmall preadditive category may be understood as a ‘ring with several objects’ (see Mitchell [20]). As such, thetheories obtained by replacing rings by preadditive categories have been developed widely in the literature (see,for instance, [1], [8], [17], [18],[19],[29], [30]). In this respect, an H -category may be seen as an “ H -modulealgebra with several objects”. Likewise, a co- H -category may be seen as an “ H -comodule algebra with severalobjects.”The various aspects of categorified Hopf actions and coactions on algebras have already been studied by severalauthors. In [13], Herscovich and Solotar obtained a Grothendieck spectral sequence for the Hochschild-Mitchell ∗ [email protected] † MB was supported by SERB Fellowship PDF/2017/000229 ‡ [email protected] § AB was partially supported by SERB Matrics fellowship MTR/2017/000112 ¶ [email protected] H -comodule category appearing as an H -Galois extension. Hopf comodule categories werealso studied in [24], where the authors introduced cleft H -comodule categories and extended classical results oncleft comodule algebras. More recently, Batista, Caenepeel and Vercruysse have shown in [2] that several deeptheorems on Hopf modules can be extended to a categorification of Hopf algebras (see also [5]).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 C . We will also construct a spectral sequence that givesthe higher derived Hom functors for relative (D , H ) -modules, where D is a co- H -category. We will developthese cohomology theories in a manner analogous to the “ H -finite cohomology” obtained by Gu´ed´enon [11] (seealso [10]) and the cohomology of relative Hopf modules studied by Caenepeel and Gu´ed´enon in [4] respectively.We now describe the paper in more detail. We begin in Section 2 by recalling the notion of a left H -categoryand a right co- H -category. For a left H -category C , we have a category of H -invariants which will be denotedby C H . For a right co- H -category D , there is a corresponding category of H -coinvariants which will be denotedby D coH . If H is a finite dimensional Hopf algebra and H ∗ is its linear dual, then a K -linear category D is aleft H ∗ -category if and only if it is a right co- H -category. In that case, D H ∗ = D coH .In Sections 3 and 4, we work with a left H -category C . We consider right C -modules that are equipped withan additional left H -equivariant structure (see Definition 3.2). This category is denoted by ( M od - C) HH . If M , N ∈ (
M od - C) HH , the space Hom
Mod - C (M , N ) of right C -module morphisms carries a left H -module structurewhose H -invariants are given by Hom
Mod - C (M , N ) H = Hom ( Mod - C) HH (M , N ) .More precisely, let ( M od - C) H denote the category with the same objects as ( M od - C) HH but whose morphismsare ordinary C -modules morphisms. Then, we show that ( M od - C) H is a left H -category and ( M od - C) HH maybe recovered as the category of H -invariants of ( M od - C) H . Further, we obtain that ( M od - C) HH is identical tothe category M od - (C H ) of right modules over the smash product category C H . In particular, this showsthat ( M od - C) HH is a Grothendieck category. We then construct a Grothendieck spectral sequence (see Theorem3.15) R p (−) H ( Ext qMod - C (M , N )) ⇒ ( R p + q Hom
Mod - (C H ) (M , −)) (N ) for the higher derived Hom in M od - (C H ) in terms of the derived Hom in M od - C and the derived functor of H -invariants.We proceed in Section 4 to develop the “ H -finite cohomology” of (C H ) -modules in a manner analogous toGu´ed´enon [11]. If M is an H -module, we denote by M ( H ) the collection of all elements m ∈ M such that Hm isa 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 L Mod - C ∶ ( M od - (C H )) op × M od - (C H ) Ð → H - mod L Mod - C (M , N ) ∶=
Hom
Mod - C (M , N ) ( H ) We then construct a Grothendieck spectral sequence (see Theorem 4.2) R p (−) ( H ) ( Ext qMod - C (M , N )) ⇒ ( R p + q L Mod - C (M , −)) (N ) The left H -category C is said to be locally finite if every morphism space Hom C ( X, Y ) is locally finite as an H -module. We denote by mod - (C H ) the full subcategory of M od - (C H ) consisting of those left H -equivariantright C -modules M such that M( X ) is H -locally finite for each X ∈ Ob (C) . When C is left H -locally finite andright noetherian, we construct a spectral sequence (see Theorem 4.19) R p (−) H ( Ext qMod - C (M , N )) ⇒ ( R p + q Hom mod - (C H ) (M , −)) (N ) In Section 5, we work with a right co- H -category D and introduce the category D M H of relative (D , H ) -Hopfmodules (see Definition 5.1). A relative (D , H ) -module consists of an H -coaction on a pair (D , M) , where2 is a left D -module. In particular, M( X ) is equipped with the structure of a right H -comodule for each X ∈ Ob (D) . We show that D M H is a Grothendieck category.Let Comod - H be the category of H -comodules. Thereafter, we construct a functor (see (5.3)) HOM D - Mod ∶ ( D M H ) op × D M H Ð→ Comod - H by using the right adjoint of the functor N ⊗ (−) ∶
Comod - H Ð→ D M H for each fixed N ∈ D M H . In the caseof an H -comodule algebra as considered by Caenepeel and Gu´ed´enon, the HOM functor gives the collectionof “rational morphisms” between relative Hopf modules (see[4, § D M H is notnecessarily enriched over Comod - H , we see that HOM D - Mod (M , N ) behaves like a
Hom object. The morphismsin
Hom D M H (M , N ) may be recovered as the H -coinvariants HOM D - Mod (M , N ) coH = Hom D M H (M , N ) . Wethen construct a Grothendieck spectral sequence (see Theorem 5.9) R p (−) coH ( R q HOM D - Mod (M , −)(N )) ⇒ ( R p + q Hom D M H (M , −)) (N ) For M , N ∈ D M H with M finitely generated as a D -module, we show that Hom D - Mod (M , N ) is an H -comodule and that HOM D - Mod (M , N ) =
Hom D - Mod (M , N ) . When D is also left noetherian, we construct aGrothendieck spectral sequence (see Theorem 5.17) R p (−) coH ( Ext q D - Mod (M , N )) ⇒ ( R p + q Hom D M H (M , −)) (N ) for the higher derived Hom in D M H . Notations:
Throughout the paper, K is a field, H is a Hopf algebra with comultiplication ∆, counit ε andbijective antipode S . We shall use Sweedler’s notation for the coproduct ∆ ( h ) = ∑ h ⊗ h and for a coaction ρ ∶ M Ð→ M ⊗ H , ρ ( m ) = ∑ m ⊗ m . We denote by H ∗ the linear dual of H . The category of left H -modules willbe denoted by H - M od and the category of right H -comodules will be denoted by Comod - H . For M ∈ H - M od ,we set M H ∶= { m ∈ M ∣ hm = ε ( h ) m ∀ h ∈ H } . For M ∈ Comod - H , we set M coH ∶= { m ∈ M ∣ ρ ( m ) = m ⊗ H } . H -categories and co- H -categories Let H be a Hopf algebra over a field K . Then, it is well known (see, for instance, [23, § H -modules as well as the category of H -comodules is monoidal. A K -linear category is said to be an H -modulecategory (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, [3, Chapter 6] or [16]. Definition 2.1. (see Cibils and Solotar [6, Definition 2.1] ) Let K be a field. A K -linear category C is said tobe a left H -module category if it is enriched over the monoidal category of left H -modules. In other words, itsatisfies the following conditions:(i) Hom C ( X, Y ) is a left H -module for all X, Y ∈ Ob (C) .(ii) h ( id X ) = ε ( h ) ⋅ id X for every X ∈ Ob (C) and every h ∈ H .(iii) The composition of morphisms in C is H -equivariant, i.e., for any h ∈ H and any pair of composablemorphisms g ∶ X Ð→ Y , f ∶ Y Ð→ Z , we have h ( f g ) = ∑ h ( f ) h ( g ) By a left H -category, we will always mean a small left H -module category. A right H -category may be definedsimilarly. efinition 2.2. Let C be a left H -module category. A morphism f ∈ Hom C ( X, Y ) is said to be H -invariant if h ( f ) = ε ( h ) ⋅ f for all h ∈ H . The subcategory whose objects are the same as those of C and whose morphismsare the H -invariant morphisms in C is denoted by 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 ) = ∑ h ( m ) h ( a ) , for all h ∈ H, a ∈ A and m ∈ M. Example 2.3. (see [15] ) 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 ∈ Ob ( H M A ) , the morphism space Hom A ( X, Y ) is a left H -module via h ( f )( x ) = ∑ h f ( S ( h ) x ) ∀ x ∈ X, ∀ f ∈ Hom A ( X, Y ) (ii) The finitely generated free right A -modules are automatically left H -equivariant. The category of (isomor-phism 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 [24]). This notionalso appears implicitly in [6]. Definition 2.4.
By a right co- H -category, we will mean a small K -linear category D that is enriched over themonoidal category of right H -comodules. In other words, we have:(i) Hom D ( X, Y ) is a right H -comodule for all X, Y ∈ Ob (D) , with structure map ρ XY ∶ Hom D ( X, Y ) Ð→ Hom D ( X, Y ) ⊗ H, ρ XY ( f ) = ∑ f ⊗ f (ii) ρ XX ( id X ) = id X ⊗ H , for any X ∈ Ob (D) and any h ∈ H .(iii) The composition of morphisms in D is H -coequivariant, i.e., for any pair of composable morphisms g ∶ X Ð→ Y , f ∶ Y Ð→ Z , we have ρ XZ ( f g ) = ∑ ( f g ) ⊗ ( f g ) = ∑ f g ⊗ f g = ρ Y Z ( f ) ρ XY ( g ) A left co- H -category may be defined similarly.A morphism f ∈ Hom D ( X, Y ) in a right co- H -category is said to be H -coinvariant if it satisfies ρ XY ( f ) = f ⊗ H .The subcategory whose objects are the same as those of D and whose morphisms are H -coinvariant is denotedby D coH . Proposition 2.5.
Let H be a finite dimensional Hopf algebra and let D be a small K -linear category. Then, D is a right co- H -category if and only if D is a left H ∗ -category. Moreover, D H ∗ = D coH .Proof. Let { e , . . . , e n } be a basis of H and let { e ∗ , . . . , e ∗ n } be its dual basis. If D is a right co- H -category, then D becomes a left H ∗ -category with h ∗ ( f ) ∶ = ∑ f h ∗ ( f ) h ∗ ∈ H ∗ and f ∈ Hom D ( X, Y ) . Indeed, it is easy to check that this action makes Hom D ( X, Y ) a left H ∗ -module for every X, Y ∈ Ob (D) and that h ∗ ( f g ) = ∑( f g ) h ∗ (( f g ) ) = ∑ f g h ∗ ( f g ) = ∑ f g h ∗ ( f ) h ∗ ( g ) = ∑ f h ∗ ( f ) g h ∗ ( g ) = ∑ h ∗ ( f ) h ∗ ( g ) . Conversely, if D is a left H ∗ -category, then D is a right co- H -category with ρ XY ∶ Hom D ( X, Y ) Ð→ Hom D ( X, Y ) ⊗ H, ρ XY ( f ) ∶ = n ∑ i = e ∗ i ( f ) ⊗ e i It may be verified that this gives a right H -comodule structure on Hom D ( X, Y ) . We need to check that thecomposition of morphisms in D is H -coequivariant. For any h ∗ ∈ H ∗ , g ∈ Hom D ( X, Y ) and f ∈ Hom D ( Y, Z ) ,we have ( id ⊗ h ∗ )( ρ XZ ( f g )) = ( id ⊗ h ∗ ) (∑ ni = e ∗ i ( f g ) ⊗ e i ) = ∑ ni = e ∗ i ( f g ) ⊗ h ∗ ( e i )= ∑ ni = ( h ∗ ( e i ) e ∗ i )( f g ) ⊗ H = h ∗ ( f g ) ⊗ H = ∑ ni = h ∗ ( f ) h ∗ ( g ) ⊗ H = ∑ ≤ i,j ≤ n ( h ∗ ( e i ) e ∗ i )( f )( h ∗ ( e j ) e ∗ j )( g ) ⊗ H = ∑ ≤ i,j ≤ n e ∗ i ( f ) e ∗ j ( g ) ⊗ h ∗ ( e i e j )= ( id ⊗ h ∗ ) (∑ ≤ i,j ≤ n e ∗ i ( f ) e ∗ j ( g ) ⊗ e i e j ) Since H is finite dimensional, it follows that ρ XZ ( f g ) = ∑ ≤ i,j ≤ n e ∗ i ( f ) e ∗ j ( g ) ⊗ e i e j = ρ Y Z ( f ) ρ XY ( g ) We also have
Hom D H ∗ ( X, Y ) = { f ∈ Hom D ( X, Y ) ∣ h ∗ ( f ) = ε H ∗ ( h ∗ ) f = h ∗ ( H ) f, ∀ h ∗ ∈ H ∗ }= { f ∈ Hom D ( X, Y ) ∣ ∑ f h ∗ ( f ) = h ∗ ( H ) f, ∀ h ∗ ∈ H ∗ }= { f ∈ Hom D ( X, Y ) ∣ ( id ⊗ h ∗ )( ρ XY ( f )) = ( id ⊗ h ∗ )( f ⊗ H ) , ∀ h ∗ ∈ H ∗ }= { f ∈ Hom D ( X, Y ) ∣ ρ XY ( f ) = f ⊗ H } = Hom D coH ( X, Y ) . 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 C H , which will be recalled in the next section. H -equivariant modules and the first spectral sequence Let C be a left H -category. In this section, we will study the category of H -equivariant C -modules and computetheir higher derived Hom functors by means of a spectral sequence. We begin with the following definition (see,for instance, [21, 25]).
Definition 3.1.
Let C be a small K -linear category. A right module over C is a K -linear functor C op Ð→ V ect K ,where V ect K denotes the category of K -vector spaces. Similarly, a left module over C is a K -linear functor C Ð→ V ect K . The category of all right (resp. left) modules over C will be denoted by M od - C (resp. C - M od ). X ∈ Ob (C) , the representable functors h X ∶ = Hom C ( − , X ) and X h ∶ = Hom C ( X, − ) are examples ofright and left modules over C respectively. Unless otherwise mentioned, by a C -module we will always mean aright C -module. Definition 3.2.
Let C be a left H -category. Let M be a right C -module with a given left H -module structureon M( X ) for each X ∈ Ob (C) . Then, M is said to be a left H -equivariant right C -module if h (M( f )( m )) = ∑ M( h f )( h m ) ∀ h ∈ H, f ∈ Hom C ( X, Y ) , m ∈ M( Y ) A morphism η ∶ M Ð→ N of left H -equivariant right C -modules is a morphism η ∈ Hom
Mod - C (M , N ) such that η ( X ) ∶ M( X ) Ð→ N ( X ) is H -linear for each X ∈ Ob (C) . We will denote the category of left H -equivariantright C -modules by ( M od - C) HH .By ( M od - C) H , we will denote the category whose objects are the same as those of ( M od - C) HH , but whose mor-phisms are those of right C -modules. Lemma 3.3.
Let C be a left H -category. Given M , N ∈ (
M od - C) H , the H -module action on Hom
Mod - C (M , N ) given by ( h ⋅ η )( X )( m ) = ∑ h η ( X )( S ( h ) m ) (3.1) for η ∈ Hom
Mod - C (M , N ) , h ∈ H , X ∈ Ob (C) , m ∈ M( X ) makes ( M od - C) H a left H -category.Proof. Using the H -equivariance of M and N , it may be verified that the action in (3.1) defines a left H -modulestructure on Hom
Mod - C (M , N ) . We now consider η ∈ Hom
Mod - C (M , N ) and ν ∈ Hom
Mod - C (N , P ) . Then, wehave ∑(( h ν )( X ) ○ ( h η )( X ))( m ) = ∑( h ν )( X ) ( h η ( X )( S ( h ) m ))= ∑ h ν ( X ) ( S ( h ) h η ( X )( S ( h ) m ))= ∑ h ν ( X ) ( η ( X )( ε ( h ) S ( h ) m ))= ∑ h ν ( X ) ( η ( X )( S ( h ) m ))= ( h ⋅ ( ν ○ η ))( X )( m ) This proves the result.
Proposition 3.4.
The category ( M od - C ) HH of left H -equivariant right C -modules is identical to (( M od - C ) H ) H .Proof. Suppose that η ∈ Hom
Mod - C ( M , N ) H . We claim that η ( X ) ∶ M ( X ) Ð→ N ( X ) is H -linear for each X ∈ Ob ( C ) . For this, we observe that η ( X )( hm ) = ∑ η ( X )( ε ( h ) h m ) = ∑ ε ( h ) η ( X )( h m ) = ∑( h ⋅ η )( X )( h m )= ∑ h η ( X )( S ( h ) h m ) = hη ( X )( m ) for any h ∈ H and m ∈ M ( X ) . Conversely, if each η ( X ) ∶ M ( X ) Ð→ N ( X ) is H -linear, it is clear from thedefinition of the left H -action in (3.1) that h ⋅ η = ε ( h ) η , i.e., η ∈ Hom
Mod - C ( M , N ) H .We will now study the category ( M od - C ) HH of left H -equivariant right C -modules. In particular, one may ask if ( M od - C ) HH is an abelian category. We will show that ( M od - C ) HH is in fact a Grothendieck category. For this,we will need to consider the smash product category of C and H . Definition 3.5. (see [6, § ) Let C be a left H -category. The smash product of C and H , denoted by C H , isthe K -linear category defined by Ob ( C H ) ∶ = Ob ( C ) Hom C H ( X, Y ) ∶ = Hom C ( X, Y ) ⊗ H n element of Hom C H ( X, Y ) is a finite sum of the form ∑ g i h i , with g i ∈ Hom C ( X, Y ) and h i ∈ H . Then,the composition of morphisms in C H is determined by ( f h )( g h ′ ) = ∑ f ( h g ) ( h h ′ ) for any pair of composable morphisms g ∶ X Ð→ Y , f ∶ Y Ð→ Z in C and any h, h ′ ∈ H . Lemma 3.6.
Let M ∈ M od - ( C H ) . Then, M ( X ) has a left H -module structure for each X ∈ Ob ( C H ) givenby hm ∶ = M ( id X S ( h ))( m ) ∀ h ∈ H, m ∈ M ( X ) Further, given any morphism η ∶ M Ð→ N in M od - ( C H ) , every η ( X ) ∶ M ( X ) Ð→ N ( X ) is H -linear.Proof. For h, h ′ ∈ H and m ∈ M ( X ) , we have h ( h ′ m ) = ( M ( id X S ( h )) ○ M ( id X S ( h ′ )))( m )= M (( id X S ( h ′ ))( id X S ( h )))( m )= ∑ M ( S ( h ′ )( id X ) S ( h ′ ) S ( h ))( m )= ∑ M ( ε ( S ( h ′ ))( id X ) S ( h ′ ) S ( h ))( m ) (using Definition 2.1(ii)) = M ( id X S ( hh ′ ))( m ) (using ε ○ S = ε ) = ( hh ′ )( m ) If η ∶ M Ð→ N is a morphism in
M od - ( C H ) , it may be verified easily that each η ( X ) ∶ M ( X ) Ð→ N ( X ) is H -linear. Proposition 3.7.
Let C be a left H -category. Then, there is a one-one correspondence between left H -equivariant right C -modules and right modules over C H .Proof. For any H -equivariant C -module M , we have the object M ′ in M od - ( C H ) defined by M ′ ( X ) ∶ = M ( X ) ∀ X ∈ Ob ( C H ) , M ′ ( f h )( m ) ∶ = S − ( h ) M ( f )( m ) ∀ f h ∈ Hom C H ( Y, X ) , m ∈ M ( X ) . (3.2)For f ′ h ′ ∈ Hom C H ( Z, Y ) , f h ∈ Hom C H ( Y, X ) and m ∈ M ( X ) , we have M ′ (( f h ) ○ ( f ′ h ′ ))( m ) = ∑ M ′ ( f ( h f ′ ) h h ′ )( m )= ∑ S − ( h h ′ ) M ( f ( h f ′ ))( m )= S − ( h ′ ) ∑ S − ( h ) M ( f ( h f ′ ))( m )= S − ( h ′ ) ∑ M ( S − ( h )( f ( h f ′ )))( S − ( h ) m ) ( since M is H -equivariant )= S − ( h ′ ) ∑ M ( S − ( h )( f ) S − ( h )( h f ′ )))( S − ( h ) m )= S − ( h ′ ) ∑ M (( S − ( h )( f )) ○ f ′ ))( S − ( h ) m )= S − ( h ′ ) M ( f ′ ) (∑ M ( S − ( h ) f )( S − ( h ) m ))= S − ( h ′ ) M ( f ′ ) ( S − ( h ) M ( f )( m ))) ( since M is H -equivariant )= ( M ′ ( f ′ h ′ ) ○ M ′ ( f h ))( m ) Conversely, given any M ′ in M od - ( C H ) , we can obtain an H -equivariant C -module defined by M ( X ) ∶ = M ′ ( X ) ∀ X ∈ Ob ( C ) M ( f ) ∶ = M ′ ( f H ) ∀ f ∈ Hom C ( Y, X ) M ( X ) = M ′ ( X ) has a left H -module structure. We now check that M isindeed H -equivariant: h ( M ( f )( m )) = M ′ ( id X S ( h ))( M ′ ( f H )( m ))= M ′ ( f S ( h ))( m )= ∑ M ′ (( id X S ( h ))( h f H ))( m )= ∑ M ′ ( h f H )( M ′ ( id X S ( h ))( m ))= ∑ M ( h f )( h m ) . Proposition 3.8.
Let M and N be right C H -modules. Then, Hom
Mod - C ( M , N ) is a left H -module and itsinvariants are given by Hom
Mod - C ( M , N ) H = Hom
Mod - (C H ) ( M , N ) .Proof. We have shown in Proposition 3.7 that every right C H -module is also a left H -equivariant right C -module. Accordingly, we use (3.1) to give an H -module structure on Hom
Mod - C ( M , N ) by setting ( h ⋅ η )( X )( m ) ∶ = ∑ h ( η ( X )( S ( h ) m )) ∀ h ∈ H, X ∈ Ob ( C ) , m ∈ M ( X ) (3.3)for any η ∈ Hom
Mod - C ( M , N ) .Suppose now that η ∈ Hom
Mod - C ( M , N ) H . From the proof of Proposition 3.4, it follows that η ( X ) ∶ M ( X ) Ð→N ( X ) is H -linear for each X ∈ Ob ( C ) . We need to show that η ∈ Hom
Mod - (C H ) ( M , N ) . For any f ∶ Y Ð→ X in C , h ∈ H and m ∈ M ( X ) , we have η ( Y )( M ( f h )( m )) = η ( Y )( S − ( h ) M ( f )( m ))= η ( Y ) ( M ( S − ( h ) f )( S − ( h ) m )) ( since M is H -equivariant )= N ( S − ( h ) f ) η ( X )( S − ( h ) m )= N ( S − ( h ) f )( S − ( h ) η ( X )( m ))= S − ( h )( N ( f )( η ( X )( m ))) ( since N is H -equivariant )= N ( f h )( η ( X )( m )) Conversely, let η ∈ Hom
Mod - (C H ) ( M , N ) . Using the H -linearity of η ( X ) from Lemma 3.6, it is clear from(3.3) that η ∈ Hom
Mod - C ( M , N ) H . Proposition 3.9.
Let C be a left H -category. Then, the categories M od - ( C H ) and ( M od - C ) HH are identical.In particular, the category ( M od - C ) HH of left H -equivariant right C -modules is a Grothendieck category.Proof. The fact that
M od - ( C H ) and ( M od - C ) HH are identical follows from Propositions 3.4, 3.7 and 3.8.Further, given any small preadditive category E , it is well known that the category M od - E is a Grothendieckcategory (see, for instance, [25, Example V.2.2]). Since C H is a small preadditive category, the result follows.We denote by M ⊗ C ( C H ) the extension of a right C -module M to a right ( C H ) -module. For the generalnotion of extension and restriction of scalars in the case of modules over a category, see, for instance, [22, § Lemma 3.10.
Let M be a right C -module. Then,(1) A right ( C H ) -module M ⊗ H may be obtained by setting ( M ⊗ H )( X ) ∶ = M ( X ) ⊗ H (( M ⊗ H )( f ′ h ′ )) ( m ⊗ h ) ∶ = ∑ M ( h f ′ )( m ) ⊗ h h ′ or any X ∈ Ob ( C H ) , f ′ h ′ ∈ Hom C H ( Y, X ) , m ∈ M ( X ) and h, h ′ ∈ H .(2) M ⊗ H is isomorphic to M ⊗ C ( C H ) as objects in M od - ( C H ) .Proof. (1) For any f ′′ h ′′ ∈ Hom C H ( Z, Y ) , f ′ h ′ ∈ Hom C H ( Y, X ) , we have (( M ⊗ H ) (( f ′ h ′ )( f ′′ h ′′ ))) ( m ⊗ h ) = ∑ (( M ⊗ H )( f ′ ( h ′ f ′′ ) h ′ h ′′ )) ( m ⊗ h )= ∑ M ( h ( f ′ ( h ′ f ′′ )) ( m ) ⊗ h h ′ h ′′ = ∑ M (( h f ′ )( h h ′ f ′′ )) ( m ) ⊗ h h ′ h ′′ ( since C is a left H -category )= ∑ M ( h h ′ f ′′ ) M ( h f ′ )( m ) ⊗ h h ′ h ′′ = ∑ M (( h h ′ ) f ′′ ) M ( h f ′ )( m ) ⊗ ( h h ′ ) h ′′ = (( M ⊗ H )( f ′′ h ′′ )( M ⊗ H )( f ′ h ′ )) ( m ⊗ h ) Further, (( M ⊗ H )( id X H )) ( m ⊗ h ) = M ( h id X )( m ) ⊗ h = m ⊗ h . Thus, M ⊗ H ∈ M od - ( C H ) .(2) It may be easily checked that the assignment M ↦ M ⊗ H defines a functor from M od - C to M od - ( C H ) ,which we denote by ( − ) ⊗ H . We will now show that the functor ( − ) ⊗ H ∶ M od - C Ð→
M od - ( C H ) is theleft adjoint to the restriction of scalars from M od - ( C H ) to M od - C , i.e., there is a natural isomorphism Hom
Mod - (C H ) ( M ⊗ H, N ) ≅ Hom
Mod - C ( M , N ) . The result of ( ) will then follow from the uniqueness ofadjoints. We define φ ∶ Hom
Mod - (C H ) ( M ⊗ H, N ) Ð→ Hom
Mod - C ( M , N ) by setting φ ( η )( X )( m ) ∶ = η ( X )( m ⊗ H ) for any η ∈ Hom
Mod - (C H ) ( M ⊗ H, N ) , X ∈ Ob ( C ) and m ∈ M ( X ) .For any f ∈ Hom C ( X, Y ) and m ′ ∈ M ( Y ) , we have N ( f ) φ ( η )( Y )( m ′ ) = N ( f H ) η ( Y )( m ′ ⊗ H ) = ( η ( X )( M ⊗ H )( f H )) ( m ′ ⊗ H )= η ( X )( M ( f )( m ′ ) ⊗ H ) = φ ( η )( X )( M ( f )( m ′ )) Thus, φ ( η ) ∈ Hom
Mod - C ( M , N ) . We now check the injectivity of φ . Let η, ν ∈ Hom
Mod - (C H ) ( M ⊗ H, N ) besuch that φ ( η ) = φ ( ν ) . Then, we have η ( X )( m ⊗ h ) = η ( X )(( M ⊗ H )( id X h ))( m ⊗ H ) = N ( id X h ) η ( X )( m ⊗ H ) = N ( id X h ) φ ( η )( X )( m )= N ( id X h ) φ ( ν )( X )( m )= ν ( X )( m ⊗ h ) for all X ∈ Ob ( C ) and m ⊗ h ∈ M ( X ) ⊗ H . This shows that η = ν . Next, given any ξ ∈ Hom
Mod - C ( M , N ) , wedefine η ( X ) ∶ M ( X ) ⊗ H Ð→ N ( X ) by η ( X )( m ⊗ h ) ∶ = S − ( h ) ξ ( X )( m ) for X ∈ Ob ( C H ) and m ⊗ h ∈ M ( X ) ⊗ H . Then, for any f ′ h ′ ∈ Hom
Mod - (C H ) ( Y, X ) , we have η ( Y ) (( M ⊗ H )( f ′ h ′ )( m ⊗ h )) = ∑ η ( Y ) ( M ( h f ′ )( m ) ⊗ h h ′ )= ∑ S − ( h h ′ ) ( ξ ( Y )( M ( h f ′ )( m )))= ∑ S − ( h h ′ ) ( N ( h f ′ ) ξ ( X )( m ))= ∑ N (( S − ( h h ′ )) h f ′ ) (( S − ( h h ′ )) ξ ( X )( m )) ( since N is H -equivariant )= ∑ N ( S − ( h ′ ) S − ( h ) h f ′ ) ( S − ( h ′ ) S − ( h ) ξ ( X )( m ))= ∑ N ( S − ( h ′ ) f ′ ) ( S − ( h ′ ) S − ( h ) ξ ( X )( m ))= S − ( h ′ ) N ( f ′ ) ( S − ( h ) ξ ( X )( m )) ( since N is H -equivariant )= N ( f ′ h ′ ) ( S − ( h ) ξ ( X )( m )) ( using (3.2) )= N ( f ′ h ′ )( η ( X )( m ⊗ h )) Hence, η ∈ Hom
Mod - (C H ) ( M ⊗ H, N ) . We also have φ ( η )( X )( m ) = η ( X )( m ⊗ H ) = ξ ( X )( m ) , i.e., φ ( η ) = ξ .Hence, φ is surjective. This proves the result. 9 roposition 3.11. (1) The extension of scalars from M od - C to M od - ( C H ) is exact.(2) Let I be an injective object in M od - ( C H ) . Then, I is also an injective object in M od - C .Proof. Let M , N ∈ M od - C be such that φ ∶ M Ð→ N is a monomorphism, i.e., φ ( X ) ∶ M ( X ) Ð→ N ( X ) is amonomorphism in V ect K for each X ∈ Ob ( C ) . Applying the isomorphism in Lemma 3.10, ( φ ⊗ C ( C H ))( X ) =( φ ⊗ H )( X ) ∶ M ( X ) ⊗ H Ð→ N ( X ) ⊗ H is a monomorphism. Since extension of scalars is a left adjoint, italready preserves colimits. This proves (1). The result of (2) now follows from [27, Tag 015Y]. Lemma 3.12.
Let M ∈ H - M od and let N ∈ M od - ( C H ) . Then, a right ( C H ) -module M ⊗ N can be definedby setting ( M ⊗ N )( X ) ∶ = M ⊗ N ( X ) ( M ⊗ N )( f )( m ⊗ n ) ∶ = m ⊗ N ( f )( n ) for any X ∈ Ob ( C ) , f ∈ Hom C ( Y, X ) and m ⊗ n ∈ M ⊗ N ( X ) .Proof. It is clear that M ⊗ N ∈ M od - C . Now for each X ∈ Ob ( C ) , the K -vector space M ⊗ N ( X ) has a left H -module structure given by h ( m ⊗ n ) ∶ = ∑ h m ⊗ h n ∀ h ∈ H It may be easily verified that M ⊗ N is an H -equivariant right C -module under this action. Therefore, M ⊗ N ∈ M od - ( C H ) by Proposition 3.7.Given any N ∈ M od - ( C H ) , let ( − ) ⊗ N ∶ H - M od Ð→ M od - ( C H ) denote the functor which takes any M ∈ H - M od to M ⊗ N ∈ M od - ( C H ) . Proposition 3.13.
Let N , P ∈ M od - ( C H ) and let M ∈ H - M od . Then, we have a natural isomorphism φ ∶ Hom
Mod - (C H ) ( M ⊗ N , P ) Ð→ Hom H - Mod ( M, Hom
Mod - C ( N , P )) given by ( φ ( η )( m ))( X )( n ) ∶ = η ( X )( m ⊗ n ) for each X ∈ Ob ( C ) and m ∈ M, n ∈ N ( X ) .Proof. Let η ∈ Hom
Mod - (C H ) ( M ⊗ N , P ) . It may be checked that φ ( η )( m ) ∈ Hom
Mod - C ( N , P ) for every m ∈ M . We now verify that φ ( η ) is H -linear, i.e., for h ∈ H : ( h ( φ ( η )( m )))( X )( n ) = ∑ h ( φ ( η )( m )( X )( S ( h ) n )) ( using Proposition 3 . )= ∑ h ( η ( X )( m ⊗ S ( h ) n ))= ∑ η ( X ) ( h m ⊗ h S ( h ) n ) ( since η ( X ) is H -linear by Lemma 3.6 )= η ( X )( hm ⊗ n ) = ( φ ( η )( hm ))( X )( n ) Clearly, φ is injective. For f ∈ Hom H - Mod ( M, Hom
Mod - C ( N , P )) , we consider ν ∈ Hom
Mod - (C H ) ( M ⊗ N , P ) determined by ν ( X )( m ⊗ n ) ∶ = f ( m )( X )( n ) (3.4)for each X ∈ Ob ( C ) , n ∈ N ( X ) and m ∈ M . We first check that ν ( X ) ∶ M ⊗ N ( X ) Ð→ P ( X ) is H -linear forevery X ∈ Ob ( C ) , i.e., for h ∈ H : ν ( X ) ( h ( m ⊗ n )) = ∑ ν ( X )( h m ⊗ h n )= ∑ f ( h m )( X )( h n )= ∑( h f ( m ))( X )( h n ) ( since f is H -linear )= ∑ h ( f ( m )( X )) ( S ( h ) h n )= h ( ν ( X )( m ⊗ n )) Using the fact that f ( m ) ∈ Hom
Mod - C ( N , P ) for each m ∈ M , it may now be verified that ν ∈ Hom
Mod - C ( M ⊗ N , P ) . From the equivalence of categories in Proposition 3.9, it follows that ν ∈ Hom
Mod - (C H ) ( M ⊗ N , P ) .From (3.4), it is also clear that φ ( ν ) = f . 10 orollary 3.14. If I is an injective object in M od - ( C H ) , then Hom
Mod - C ( N , I ) is an injective object in H - M od for any N ∈ M od - ( C H ) .Proof. From Proposition 3.13, we know that the functor ( − ) ⊗ N ∶ H - M od Ð→ M od - ( C H ) is a left adjointand therefore preserves colimits. Further, given a monomorphism M ↪ M in H - M od , it is clear from thedefinition in Lemma 3.12 that M ⊗ N Ð→ M ⊗ N is a monomorphism in M od - ( C H ) . Hence, ( − ) ⊗ N ∶ H - M od Ð→ M od - ( C H ) is exact. As such, its right adjoint Hom
Mod - C ( N , ) ∶ M od - ( C H ) Ð→ H - M od preservesinjectives.We denote by ( − ) H the functor from H - M od to V ect K that takes M to M H = { m ∈ M ∣ hm = ε ( h ) m ∀ h ∈ H } .We now recall from Proposition 3.8 that we have an isomorphism Hom
Mod - C ( M , N ) H ≅ Hom
Mod - (C H ) ( M , N ) for any M , N ∈ M od - ( C H ) . At the level of the derived Hom functors, this leads to the following spectralsequence.
Theorem 3.15.
Let M , N ∈ M od - ( C H ) . Then, there exists a first quadrant spectral sequence: R p ( − ) H ( Ext qMod - C ( M , N )) ⇒ ( R p + q Hom
Mod - (C H ) ( M , − )) ( N ) Proof.
We consider the functors F ∶ = Hom
Mod - C ( M , − ) ∶ M od - ( C H ) Ð→ H - M od and G ∶ = ( − ) H ∶ H - M od Ð→ V ect K . We notice that M od - ( C H ) , H - M od and
V ect K are all Grothendieck categories. From Corollary 3.14,we know that F preserves injectives. Using Proposition 3.8, we see that the functor ( G ○ F ) ∶ M od - ( C H ) Ð→ V ect K is given by ( G ○ F )( N ) = Hom
Mod - C ( M , N ) H = Hom
Mod - (C H ) ( M , N ) . The result now follows fromthe Grothendieck spectral sequence for composite functors (see [9]). H -locally finite modules and cohomology We recall the definition of H -locally finite modules from [11]. For M ∈ H - M od and m ∈ M , let Hm be the H -submodule of M spanned by the elements hm for h ∈ H . Consider M ( H ) ∶ = { m ∈ M ∣ Hm is finite dimensional as a K -vector space } 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 fullsubcategory of H - M od whose objects are H -locally finite H -modules will be denoted by H - mod .By Proposition 3.8, Hom
Mod - C ( N , P ) is an H -module for any N , P ∈ M od - ( C H ) . We set L Mod - C ( N , P ) ∶ = Hom
Mod - C ( N , P ) ( H ) Clearly, this defines a functor L Mod - C ( N , − ) ∶ M od - ( C H ) Ð→ H - mod for every N ∈ M od - ( C H ) . Proposition 4.1.
Let N ∈ M od - ( C H ) . Then, the functor L Mod - C ( N , − ) ∶ M od - ( C H ) Ð→ H - mod is rightadjoint to the functor ( − ) ⊗ N ∶ H - mod Ð→ M od - ( C H ) , i.e., we have natural isomorphisms Hom
Mod - (C H ) ( M ⊗ N , P ) ≅ Hom H - mod ( M, L Mod - C ( N , P )) for all P ∈ M od - ( C H ) and M ∈ H - mod . roof. Let φ ∶ Hom
Mod - (C H ) ( M ⊗ N , P ) Ð→ Hom H - Mod ( M, Hom
Mod - C ( N , P )) be the isomorphism as inProposition 3.13. Let η ∶ M ⊗ N Ð→ P be a morphism in
M od - ( C H ) . It follows that Hφ ( η )( m ) is finitedimensional for each m ∈ M by observing that φ ( η ) is H -linear and that M is H -locally finite. Since H - mod is a full subcategory of H - M od , we have
Hom
Mod - (C H ) ( M ⊗ N , P ) ≅ Hom H - Mod ( M, L Mod - C ( N , P )) ≅ Hom H - mod ( M, L Mod - C ( N , P )) . For any M ∈ M od - ( C H ) , we can now consider the functor L Mod - C ( M , − ) ∶ M od - ( C H ) Ð→ H - mod N ↦ L
Mod - C ( M , N ) (4.1)Since M od - ( C H ) is a Grothendieck category, we obtain derived functors R p L Mod - C ( M , − ) ∶ M od - ( C H ) Ð→ H - mod , p ≥
0. We use the boldface notation to distinguish these from the functors R p L Mod - C ( M , − ) that willappear later in the proof of Proposition 4.18 as derived functors of a restriction of L Mod - C ( M , − ) . Theorem 4.2.
Let M , N ∈ M od - ( C H ) . We consider the functors F = Hom
Mod - C ( M , − ) ∶ M od - ( C H ) Ð→ H - M od
N ↦
Hom
Mod - C ( M , N ) G = ( − ) ( H ) ∶ H - M od Ð→ H - mod M ↦ M ( H ) Then, we have the following spectral sequence R p ( − ) ( H ) ( Ext qMod - C ( M , N )) ⇒ ( R p + q L Mod - C ( M , − )) ( N ) Proof.
We have ( G ○ F )( N ) = Hom
Mod - C ( M , N ) ( H ) = L Mod - C ( M , N ) . By definition, R q F ( N ) = H q ( F ( I ∗ )) = H q ( Hom
Mod - C ( M , I ∗ )) where I ∗ is an injective resolution of N in M od - ( C H ) . By Corollary 3.11, injectives in M od - ( C H ) are alsoinjectives in M od - C . Hence, R q F ( N ) = Ext qMod - C ( M , N ) . For any injective I in M od - ( C H ) , we know that F ( I ) is injective in H - M od by Corollary 3.14. Since the category H - M od has enough injectives, the result nowfollows from Grothendieck spectral sequence for composite functors (see [9]).
Definition 4.3.
Let C be a left H -category.(1) C is said to be H -locally finite if the H -module Hom C ( X, Y ) is H -locally finite, i.e., Hom C ( X, Y ) ( H ) = Hom C ( X, Y ) , for all X, Y ∈ Ob ( C ) .(2) Let M ∈ M od - ( C H ) . Then, M is said to be H -locally finite if the H -module M ( X ) is H -locally finite,i.e. M ( X ) ( H ) = M ( X ) , for each X ∈ Ob ( C ) . The full subcategory of M od - ( C H ) whose objects are H -locally finite right ( C H ) -modules will be denoted by mod - ( C H ) . If M , M ′ ∈ H - M od , we know that H acts diagonally on their tensor product M ⊗ M ′ over K , i.e., h ( m ⊗ m ′ ) =∑ h m ⊗ h m ′ for h ∈ H , m ∈ M and m ′ ∈ M ′ . In particular, if M , M ′ ∈ H - mod , it follows that M ⊗ M ′ ∈ H - mod . Accordingly, if N ∈ mod - ( C H ) and M ∈ H - mod , it is clear from the definition in Lemma 3.12 that M ⊗ N ∈ mod - ( C H ) . Corollary 4.4.
Let N ∈ mod - ( C H ) . Then, the functor L Mod - C ( N , − ) ∶ mod - ( C H ) Ð→ H - mod is rightadjoint to the functor ( − ) ⊗ N ∶ H - mod Ð→ mod - ( C H ) , i.e., we have natural isomorphisms Hom mod - (C H ) ( M ⊗ N , P ) ≅ Hom H - mod ( M, L Mod - C ( N , P )) for all P ∈ mod - ( C H ) and M ∈ H - mod . roof. This follows from Proposition 4.1 because mod - ( C H ) is a full subcategory of M od - ( C H ) . Lemma 4.5.
Let C be a left H -category. Given X ∈ Ob ( C ) , consider the representable functor h X ∈ M od - C .Then, the right C -module h X is also a right ( C H ) -module.Proof. For each Y ∈ Ob ( C ) , we have h X ( Y ) = Hom C ( Y, X ) . Since C is a left H -category, h X ( Y ) has a left H -module structure. For any f ∈ Hom C ( Z, Y ) , g ∈ h X ( Y ) and h ∈ H , we have h ( h X ( f )( g )) = h ( gf ) = ∑ h ( g ) h ( f ) = ∑ h X ( h f )( h g ) Thus, h X is a left H -equivariant right C -module. Hence, h X ∈ M od - ( C H ) by Proposition 3.9. Lemma 4.6. (1) If I is an injective in M od - ( C H ) , then L Mod - C ( N , I ) is an injective in H - mod for any N ∈ M od - ( C H ) .(2) If I is an injective in mod - ( C H ) , then(i) L Mod - C ( N , I ) is an injective in H - mod for any N ∈ mod - ( C H ) .(ii) Let C be H -locally finite. Then, for each X ∈ Ob ( C H ) , I ( X ) is an injective in H - mod .Proof. (1) The functor L Mod - C ( N , − ) ∶ M od - ( C H ) Ð→ H - mod is right adjoint to the functor ( − ) ⊗ N ∶ H - mod Ð→ M od - ( C H ) by Proposition 4.1. Further, the functor ( − ) ⊗ N always preserves monomorphisms(see the proof of Corollary 3.14). The result now follows from [27, 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 Proposition4.1. To prove (ii), we consider for each X ∈ Ob ( C ) the representable functor h X ∈ M od - C . Using Lemma4.5, we know that h X ∈ M od - ( C H ) . Further, since C is H -locally finite, we see that h X ∈ mod - ( C H ) .Using (i), we have L Mod - C ( h X , I ) is injective in H -mod. Finally, by Yoneda lemma, we have L Mod - C ( h X , I ) = Hom
Mod - C ( h X , I ) ( H ) = I ( X ) ( H ) = I ( X ) . Lemma 4.7.
Let C be an H -locally finite category. Then, for any M in M od - ( C H ) , we may obtain an object M ( H ) ∈ mod - ( C H ) by setting M ( H ) ( X ) ∶ = M ( X ) ( H ) = { m ∈ M ( X ) ∣ Hm is finite dimensional } M ( H ) ( f h )( m ) ∶ = M ( f h )( m ) for any f h ∈ Hom C H ( Y, X ) and m ∈ M ( H ) ( X ) .Proof. We need to verify that M ( f h ) ∶ M ( X ) Ð→ M ( Y ) restricts to a map M ( X ) ( H ) Ð→ M ( Y ) ( H ) . Forthis, we consider m ∈ M ( X ) ( H ) . Since M ∈ M od - ( C H ) may be treated as a left H -equivariant right C -moduleas in Proposition 3.7, we obtain h ′ M ( f h )( m ) = h ′ S − ( h )( M ( f )( m )) = ∑ M ( h ′ S − ( h ) f )( h ′ S − ( h ) m ) (4.2)for any h ′ ∈ H . Since the category C is H -locally finite and m ∈ M ( H ) ( X ) , it is clear from (4.2) that M ( f h )( m ) ∈ M ( Y ) ( H ) = M ( H ) ( Y ) . This proves the result. Proposition 4.8. (1) Let ( − ) ( H ) ∶ H - M od Ð→ H - mod be the functor N ↦ N ( H ) . Then ( − ) ( H ) is right adjoint to the forgetfulfunctor from the category H - mod to the category H - M od , i.e., we have natural isomorphisms
Hom H - Mod ( M, N ) ≅
Hom H - mod ( M, N ( H ) ) for any M ∈ H - mod and N ∈ H - M od .
2) Let ( − ) ( H ) ∶ M od - ( C H ) Ð→ mod - ( C H ) be the functor N ↦ N ( H ) . Then ( − ) ( H ) is right adjoint tothe forgetful functor from the category mod - ( C H ) to the category M od - ( C H ) , i.e., we have naturalisomorphisms Hom
Mod - (C H ) ( M , N ) ≅ Hom mod - (C H ) ( M , N ( H ) ) for any M ∈ mod - ( C H ) and N ∈ M od - ( C H ) .Proof. (1) Given any M ∈ H - mod , N ∈ H - M od and an H -module morphism φ ∶ M Ð→ N , it is clear that φ ( m ) ∈ N ( H ) for all m ∈ M .(2) Let M ∈ mod - ( C H ) . By Lemma 3.6, a morphism η ∶ M Ð→ N in M od - ( C H ) induces H -linear morphisms η ( X ) ∶ M ( X ) Ð→ N ( X ) for each X ∈ Ob ( C ) . Since M ( X ) is H -locally finite, each η ( X ) can be written as amorphism M ( X ) Ð→ N ( X ) ( H ) . The result is now clear. Lemma 4.9.
Let C be H -locally finite. Then,(1) The category mod - ( C H ) is abelian.(2) If I is an injective in M od - ( C H ) , then I ( H ) is an injective in mod - ( C H ) .(3) The category mod - ( C H ) has enough injectives.Proof. (1) Since H - mod is closed under kernels and cokernels, it is clear that the subcategory mod - ( C H ) ofthe abelian category M od - ( C H ) is closed under kernels and cokernels. Also, since products and coproductsof finitely many objects M i in mod - ( C H ) are given by ( ∏ M i )( X ) = ( ∐ M i )( X ) = ⊕ i M i ( X ) for X ∈ Ob ( C H ) , it follows that finite products and coproducts exist and coincide in mod - ( C H ) . Thus, thecategory mod - ( C H ) is abelian.(2) Since the functor ( − ) ( H ) is right adjoint to the forgetful functor in Proposition 4.8(2) and the forgetfulfunctor always preserves monomorphisms, this result follows from [27, Tag 015Y].(3) Since M od - ( 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 [20, §
3] and[21]. Given M ∈ M od - C , we set el ( M ) ∶ = ∐ X ∈ Ob (C) M ( X ) to be the collection of all elements of M . If m ∈ el ( M ) lies in M ( X ) , we write ∣ m ∣ = X . Definition 4.10.
Let C be a small preadditive category and let M ∈ M od - C .(i) A family of elements { m i ∈ el ( M )} i ∈ I is said to generate M if every element y ∈ el ( M ) can be expressedas y = ∑ i ∈ I M ( f i )( m i ) for some f i ∈ Hom C (∣ y ∣ , ∣ m i ∣) , where all but a finite number of the f i are zero.Equivalently, the family { m i ∈ el ( M )} i ∈ I is said to generate M if the induced morphism η ∶ ⊕ i ∈ I h ∣ m i ∣ Ð→ M which takes ( , ..., , id ∣ m i ∣ , , ..., ) to m i is an epimorphism. The family is said to be a basis for M if η is an isomorphism. The module M is said to be finitely generated (resp. free) if it has a finite set ofgenerators (resp. a basis).(ii) The module M is called noetherian if it satisfies the ascending chain condition on submodules. Thecategory C is said to be right noetherian if h X ∈ M od - C is noetherian for each X ∈ Ob ( C ) . roposition 4.11. Let C be a left H -category. An object M ∈ mod - ( C H ) is finitely generated in M od - ( C H ) if and only if there exists a finite dimensional V ∈ H - M od and an epimorphism V ⊗ ( n ⊕ i = h X i ) Ð→ M in M od - ( C H ) for finitely many objects { X i } ≤ i ≤ n in C , where each h X i is viewed as an object in M od - ( C H ) .Proof. Let M ∈ mod - ( C H ) be finitely generated in M od - ( C H ) . We consider a finite generating family { m i ∈ el ( M )} ≤ i ≤ n for M . Since M ∈ mod - ( C H ) , each M (∣ m i ∣) is H -locally finite and hence the H -module V ∶ = n ⊕ i = Hm i is finite dimensional. For each Y ∈ Ob ( C H ) , we consider the morphism determined by setting η ( Y ) ∶ V ⊗ ( n ⊕ i = h ∣ m i ∣ ( Y )) Ð→ M ( Y ) hm i ⊗ ( f , . . . , f n ) ↦ M ( f i h )( m i ) It is easy to check that η is a morphism in M od - ( C H ) and η ( Y ) is an epimorphism for all Y ∈ Ob ( C H ) .Conversely, let { v , . . . , v k } be a basis of a finite dimensional H -module V and X , . . . , X n be finitely manyobjects in C such that there is an epimorphism η ∶ V ⊗ ( n ⊕ i = h X i ) Ð→ M in M od - ( C H ) . It may be verified that the elements { m ij = η ( X i ) ( v j ⊗ id X i ) ∈ M ( X i )} ≤ i ≤ n, ≤ j ≤ k give a familyof generators for M . Corollary 4.12.
An object M in mod - ( C H ) is finitely generated in M od - ( C H ) if and only if M is finitelygenerated in M od - C .Proof. Let M ∈ mod - ( C H ) be finitely generated in M od - C . Then, there is a finite family { m i ∈ el ( M )} i ∈ I ofelements of M such that every y ∈ el ( M ) can be expressed as y = ∑ i ∈ I M ( f i )( m i ) for some f i ∈ Hom C (∣ y ∣ , ∣ m i ∣) .Then, y = ∑ i ∈ I M ( f i H )( m i ) and hence M is finitely generated as a ( C H ) -module.Conversely, let M in mod - ( C H ) be finitely generated in M od - ( C H ) and let η ∶ V ⊗ ( n ⊕ i = h X i ) Ð→ M (4.3)denote the epimorphism in
M od - ( C H ) as in Proposition 4.11. In particular, η is an epimorphism in M od - C .Then, if { v , ..., v k } is a basis for V , it follows from the epimorphism in (4.3) that { m ij = η ( X i )( v j ⊗ id X i ) ∈ M ( X i )} ≤ i ≤ n, ≤ j ≤ k gives a finite set of generators for M as a right C -module.We remark here that if V and V ′ are left H -modules, then Hom K ( V, V ′ ) carries a left H -module action definedby ( hf )( v ) = ∑ h f ( S ( h ) v ) ∀ v ∈ V, h ∈ H (4.4)This may be seen as the special case of the action described in Proposition 3.8 when C is the category with oneobject having endomorphism ring K . Lemma 4.13.
Let V ∈ H - M od and N ∈ M od - ( C H ) . Then, Hom K ( V, N ( X )) and Hom
Mod - C ( V ⊗ h X , N ) are isomorphic as objects in H - M od for each X ∈ Ob ( C ) . roof. We check that the canonical isomorphism φ ∶ Hom
Mod - C ( V ⊗ h X , N ) Ð→ Hom K ( V, Hom
Mod - C ( h X , N )) ≅ Hom K ( V, N ( X )) defined by φ ( η )( v ) ∶ = η ( X )( v ⊗ id X ) for any morphism η ∈ Hom
Mod - C ( V ⊗ h X , N ) and v ∈ V is H -linear: φ ( hη )( v ) = ( hη )( X )( v ⊗ id X )= ∑ h η ( X )( S ( h )( v ⊗ id X ))= ∑ h η ( X )( S ( h ) v ⊗ S ( h ) id X )= ∑ h η ( X )( S ( h ) v ⊗ ε ( S ( h )) id X )= ∑ h η ( X )( S ( h ) v ⊗ ε ( h ) id X )= ∑ h η ( X )( S ( h ) v ⊗ id X )= ∑ h φ ( η )( S ( h ) v ) = ( hφ ( η ))( v ) Proposition 4.14.
Let M and N be in mod - ( C H ) with M finitely generated in M od - ( C H ) . Then, Hom
Mod - C ( M , N ) is H -locally finite, i.e., L Mod - C ( M , N ) = Hom
Mod - C ( M , N ) ( H ) = Hom
Mod - C ( M , N ) .Proof. Since M ∈ mod - ( C H ) is finitely generated in M od - ( C H ) , there exists by Proposition 4.11 a finitedimensional H -module V and an epimorphism ϕ ∶ V ⊗ ( ⊕ ni = h X i ) Ð→ M in M od - ( C H ) for finitely manyobjects X , . . . , X n in C . Thus we get a monomorphismˆ ϕ ∶ Hom
Mod - C ( M , N ) Ð→ Hom
Mod - C ( V ⊗ ( n ⊕ i = h X i ) , N ) , η ↦ η ○ ϕ. (4.5)For each X ∈ Ob ( C ) , v ∈ V and f ∈ h X i ( X ) for some chosen 1 ≤ i ≤ n , we haveˆ ϕ ( hη )( X )( v ⊗ f ) = ( hη ○ ϕ )( X )( v ⊗ f ) = ( hη )( X ) ϕ ( X )( v ⊗ f ) = ∑ h η ( X )( S ( h ) ϕ ( X )( v ⊗ f ))= ∑ h η ( X )( ϕ ( X )( S ( h )( v ⊗ f ))) = ∑ h ( η ○ ϕ )( X )( S ( h )( v ⊗ f ))= ( h ( η ○ ϕ ))( X )( v ⊗ f ) = ( h ˆ ϕ ( η ))( X )( v ⊗ f ) This shows that ˆ ϕ is an H -module monomorphism. By Lemma 4.13, we know that Hom
Mod - C ( V ⊗ ( n ⊕ i = h X i ) , N ) ≅ n ⊕ i = Hom K ( V, N ( X i )) as H -modules. Since V is finite dimensional, we know that Hom K ( V, N ( X i )) ≅ N ( X i ) ⊗ V ∗ in V ect K and it iseasily seen that this is an isomorphism of H -modules. Since N is an H -locally finite ( C H ) -module and V ∗ is H -locally finite (because dim K ( V ∗ ) < ∞ ), it follows that each N ( X i ) ⊗ V ∗ is H -locally finite. The embeddingin (4.5) now shows that Hom
Mod - C ( M , N ) is H -locally finite. Lemma 4.15.
Let M , N ∈ M od - ( C H ) . For a morphism η ∈ Hom
Mod - C ( M , N ) , the following are equivalent:(1) η ∈ Hom
Mod - C ( M , N ) ( H ) = L Mod - C ( M , N ) .(2) There exists a finite dimensional H -module V , an element v ∈ V and some ˆ η ∈ Hom
Mod - (C H ) ( V ⊗ M , N ) such that ˆ η ( X )( v ⊗ m ) = η ( X )( m ) for each X ∈ Ob ( C ) and m ∈ M ( X ) .Proof. (1) ⇒ (2) : We put V = Hη . Since η ∈ Hom
Mod - C ( M , N ) ( H ) , we see that V is finite dimensional. Let { η , ..., η k } be a basis for V = Hη . Any element hη ∈ V can now be expressed as hη = ∑ ki = α i ( h ) η i .16e now define ˆ η ∈ Hom
Mod - C ( V ⊗ M , N ) by setting ˆ η ( X )( hη ⊗ m ) ∶ = ( hη )( X )( m ) for each h ∈ H , X ∈ Ob ( C ) and m ∈ M ( X ) . It is clear that ˆ η ( X )( η ⊗ m ) = η ( X )( m ) .In order to show that ˆ η ∈ Hom
Mod - (C H ) ( V ⊗ M , N ) , it suffices to show that each ˆ η ( X ) ∶ V ⊗ M ( X ) Ð→ N ( X ) is H -linear. For h ′ ∈ H , we haveˆ η ( X )( h ′ ( hη ⊗ m )) = ∑ ˆ η ( X )( h ′ hη ⊗ h ′ m ) = ∑ ∑ ki = ˆ η ( X )( α i ( h ′ h ) η i ⊗ h ′ m )= ∑ ∑ ki = α i ( h ′ h ) ˆ η ( X )( η i ⊗ h ′ m ) = ∑ ∑ ki = α i ( h ′ h ) η i ( X )( h ′ m )= ∑( h ′ hη )( X )( h ′ m ) = ∑ h ′ h η ( X )( S ( h ) S ( h ′ ) h ′ m )= ∑ h ′ h η ( X )( S ( h ) ε ( h ′ ) m )= ∑ h ′ h η ( X )( S ( h ) m ) = ∑ h ′ (( hη )( X )( m )) = h ′ ˆ η ( X )( hη ⊗ m ) (2) ⇒ (1) : We are given ˆ η ∈ Hom
Mod - (C H ) ( V ⊗ M , N ) . Let { v , ..., v k } be a basis for V and suppose that hv = ∑ ki = α i ( h ) v i . For each 1 ≤ i ≤ k , we define ξ i ∈ Hom
Mod - C ( M , N ) by setting ξ i ( X )( m ) ∶ = ˆ η ( X )( v i ⊗ m ) for X ∈ Ob ( C ) , m ∈ M ( X ) . For any h ∈ H , we see that ( hη )( X )( m ) = ∑ h η ( X )( S ( h ) m ) = ∑ h ˆ η ( X )( v ⊗ S ( h ) m )= ∑ ˆ η ( X )( h v ⊗ h S ( h ) m ) = ˆ η ( X )( hv ⊗ m )= ∑ ki = α i ( h ) ˆ η ( X )( v i ⊗ m ) = ∑ ki = α i ( h ) ξ i ( X )( m ) It follows from the above that Hη lies in the space generated by the finite collection { ξ , ..., ξ k } ∈ Hom
Mod - C ( M , N ) .This proves the result. Proposition 4.16. If I is an injective object in mod - ( C H ) , then L Mod - C ( − , I ) is an exact functor from mod - ( C H ) to H - mod .Proof. Let 0
Ð→ M i Ð→ N Ð→ P Ð→ mod - ( C H ) and I be an injective object in mod - ( C H ) . Then, 0 Ð→ Hom
Mod - C ( P , I ) Ð→ Hom
Mod - C ( N , I ) Ð→ Hom
Mod - C ( M , I ) is an exact sequencein H - M od . Since the functor ( − ) ( H ) ∶ H - M od Ð→ H - mod is a right adjoint by Proposition 4.8(1), it preservesmonomorphisms. Thus, 0 Ð→ L
Mod - C ( P , I ) Ð→ L
Mod - C ( N , I ) Ð→ L
Mod - C ( M , I ) is an exact sequence in H - mod .Let η ∈ L Mod - C ( M , I ) . We set V ∶ = Hη . Then, V is a finite dimensional H -module and therefore V ∈ H - mod .Thus, V ⊗ M , V ⊗ N ∈ mod - ( C H ) and we have a monomorphism id V ⊗ i ∶ V ⊗ M Ð→ V ⊗ N in mod - ( C H ) .Now, we consider the morphism ζ ∈ Hom
Mod - C ( V ⊗ M , I ) defined by setting ζ ( X )( ν ⊗ m ) ∶ = ν ( X )( m ) foreach X ∈ Ob ( C ) , m ∈ M ( X ) and ν ∈ V . It may be verified that ζ ( X ) is H -linear for each X ∈ Ob ( C ) . Thus, ζ ∈ Hom mod - (C H ) ( V ⊗ M , I ) . Since I is injective in mod - ( C H ) , there exists a morphism ξ ∶ V ⊗ N Ð→ I in mod - ( C H ) such that ξ ( id V ⊗ i ) = ζ . The morphism ξ ∈ Hom
Mod - (C H ) ( V ⊗ N , I ) now induces a morphismˆ ξ ∈ Hom
Mod - C ( N , I ) defined by setting ˆ ξ ( X )( n ) ∶ = ξ ( X )( η ⊗ n ) for every X ∈ Ob ( C ) and n ∈ N ( X ) . ApplyingLemma 4.15, we see that ˆ ξ ∈ L Mod - C ( N , I ) . Also, ˆ ξ ○ i = η . This completes the proof. Proposition 4.17.
Let C be a left H -locally finite category which is right noetherian. Let M ∈ mod - ( C H ) befinitely generated as an object in M od - ( C H ) . If I is an injective object in mod - ( C H ) , then Ext pMod - C ( M , I ) = for all p > .Proof. Since M ∈ mod - ( C H ) is finitely generated in M od - ( C H ) , by Proposition 4.11, there exists a finitedimensional H -module V and an epimorphism η ∶ P ∶ = V ⊗ ( n ⊕ i = h X i ) Ð→ M , M od - ( C H ) for finitely many objects { X i } ≤ i ≤ n in C , where h X i are viewed as objects in M od - ( C H ) .Since C is H -locally finite, each h X i ∈ mod - ( C H ) . Since V is finite dimensional, we must have V ∈ H - mod .Thus, P ∈ mod - ( C H ) . Using Proposition 4.11 and Corollary 4.12, it follows that P is finitely generated in M od - C . Since C is right noetherian, P is a noetherian right C -module (see, for instance, [20, § ( C H ) -submodule K ∶ = Ker ( η ) of P is finitely generated in M od - C . So, again using Proposition 4.11 and Corollary 4.12 it follows that thereexists a finite dimensional H -module V and an epimorphism η ∶ P ∶ = V ⊗ ( n ⊕ j = h Y j ) Ð→ K , in M od - ( C H ) for finitely many objects { Y j } ≤ j ≤ n in C . Since V and V are finite dimensional K -vectorspaces, clearly P and P are free right C -modules. Moreover, Im ( η ) = K = Ker ( η ) . Thus, continuing in thisway, we can construct a free resolution of the module M in the category M od - C : P ∗ = ⋯ Ð→ P i Ð→ ⋯ Ð→ P Ð→ P Ð→ M Ð→ pMod - C ( M , I ) = H p ( Hom
Mod - C ( P ∗ , I )) , p ≥ M and { P i } i ≥ are finitely generated in M od - ( C H ) , we have L Mod - C ( M , I ) = Hom
Mod - C ( M , I ) and L Mod - C ( P i , I ) = Hom
Mod - C ( P i , I ) for every i ≥
0, by Proposition 4.14. From Proposition 4.16, we know that L Mod - C ( − , I ) is exact and it follows that H p ( L Mod - C ( P ∗ , I )) = p >
0. This proves the result.
Proposition 4.18.
Let C be a left H -locally finite category which is right noetherian. Let M and N be in mod - ( C H ) with M finitely generated in M od - ( C H ) and let E ∗ be an injective resolution of N in mod - ( C H ) .Then, Ext pMod - C ( M , N ) = H p ( Hom
Mod - C ( M , E ∗ )) , p ≥ . Proof.
Let P ∗ be the free resolution of M in M od - C constructed as in the proof of Proposition 4.17. Then, wehave Ext pMod - C ( M , N ) = H p ( Hom
Mod - C ( P ∗ , N )) = H p ( L Mod - C ( P ∗ , N )) where the second equality follows from Proposition 4.14. Since H - mod is an abelian category and L Mod - C ( P ∗ , N ) is a complex in H - mod , it follows that H p ( L Mod - C ( P ∗ , N )) ∈ H - mod . Hence, we may consider the family { Ext pMod - C ( M , − )} p ≥ as a δ -functor from mod - ( C H ) to H - mod .By Proposition 4.17, Ext pMod - C ( M , I ) = , p > I in mod - ( C H ) . Since mod - ( C H ) has enough injectives, it follows that each Ext pMod - C ( M , − ) ∶ mod - ( C H ) Ð→ H - mod is effaceable (see, forinstance, [12, § III.1]).Since mod - ( C H ) has enough injectives, we can consider the right derived functors R p L Mod - C ( M , − ) ∶ mod - ( C H ) Ð→ H - mod For p =
0, we notice that Ext Mod - C ( M , − ) = Hom
Mod - C ( M , − ) = L Mod - C ( M , − ) = R L Mod - C ( M , − ) as functorsfrom mod - ( C H ) to H - mod . Since each Ext pMod - C ( M , − ) is effaceable for p >
0, the family { Ext pMod - C ( M , − )} p ≥ forms a universal δ -functor and it follows from [12, Corollary III.1.4] thatExt pMod - C ( M , − ) = R p L Mod - C ( M , − ) ∶ mod - ( C H ) Ð→ H - mod for every p ≥
0. Therefore, we haveExt pMod - C ( M , N ) = ( R p L Mod - C ( M , − ))( N ) = H p ( L Mod - C ( M , E ∗ )) = H p ( Hom
Mod - C ( M , E ∗ )) heorem 4.19. Let C be a left H -locally finite category which is right noetherian. Fix M ∈ mod - ( C H ) with M finitely generated in M od - ( C H ) . We consider the functors F = Hom
Mod - C ( M , − ) ∶ mod - ( C H ) Ð→ H - mod N ↦
Hom
Mod - C ( M , N ) G = ( − ) H ∶ H - mod Ð→ V ect K M ↦ M H Then, we have the following spectral sequence R p ( − ) H ( Ext qMod - C ( M , N )) ⇒ ( R p + q Hom mod - (C H ) ( M , − )) ( N ) Proof.
Using Proposition 3.8 and the fact that mod - ( C H ) is a full subcategory of M od - ( C H ) , we have ( G ○ F )( N ) = Hom
Mod - C ( M , N ) H = Hom mod - (C H ) ( M , N ) . By definition, R q F ( N ) = H q ( F ( E ∗ )) = H q ( Hom
Mod - C ( M , E ∗ )) where E ∗ is an injective resolution of N in mod - ( C H ) . By Proposition 4.18, we get R q F ( N ) = Ext qMod - C ( M , N ) .For any injective I in mod - ( C H ) , we know that F ( I ) is an injective in H - mod by Proposition 4.14 andLemma 4.6(2). Since the category H - mod has enough injectives (see,[11, Lemma 1.4]), the result now followsfrom Grothendieck spectral sequence for composite functors (see [9]). (D , H ) -Hopf modules Let D be a right co- H -category. In the notation of Definition 2.4, for any X , Y ∈ Ob ( D ) , there is an H -coactionon the K -vector space Hom D ( X, Y ) given by ρ XY ( f ) ∶ = ∑ f ⊗ f . In this section, we will study the relativeHopf modules over the category 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 -coactiongiven by ρ M ∶ M Ð→ M ⊗ H , we set M coH ∶ = { m ∈ M ∣ ρ M ( m ) = m ⊗ H } to be the coinvariants of M . Definition 5.1.
Let D be a right co- H -category. Let M be a left D -module with a given right H -comodulestructure ρ M( X ) ∶ M ( X ) Ð→ M ( X ) ⊗ H , m ↦ ∑ m ⊗ m on M ( X ) for each X in Ob ( D ) . Then, M is saidto be a relative ( D , H ) -Hopf module if the following condition holds: ρ M( Y ) ( M ( f )( m )) = ∑ M ( f )( m ) ⊗ f m (5.1) for any f ∈ Hom D ( X, Y ) and m ∈ M ( X ) . We denote by D M H the category whose objects are relative ( D , H ) -Hopf modules and whose morphisms are given by Hom D M H ( M , N ) ∶ = { η ∈ Hom D - Mod ( M , N ) ∣ η ( X ) ∶ M ( X ) Ð→ N ( X ) is H -colinear ∀ X ∈ Ob ( D )} . We now recall the tensor product of H -comodules. Let M, N ∈ Comod - H with H -coactions ρ M and ρ N ,respectively. Then, M ⊗ N ∈ Comod - H with H -coaction given by ρ M ⊗ N ∶ = ( id ⊗ id ⊗ m H )( id ⊗ τ ⊗ id )( ρ M ⊗ ρ N ) ,where m H denotes the multiplication on H and id ⊗ τ ⊗ id ∶ M ⊗ ( H ⊗ N ) ⊗ H Ð→ M ⊗ ( N ⊗ H ) ⊗ H denotesthe twist map. In other words, ρ M ⊗ N ( m ⊗ n ) = ∑ m ⊗ n ⊗ m n for m ⊗ n ∈ M ⊗ N . Lemma 5.2.
Let M ∈ Comod - H and N ∈ D M H . Then, N ⊗ M defined by setting ( N ⊗ M )( X ) ∶ = N ( X ) ⊗ M ( N ⊗ M )( f )( n ⊗ m ) ∶ = N ( f )( n ) ⊗ m for each X ∈ Ob ( D ) , f ∈ Hom D ( X, Y ) and n ⊗ m ∈ N ( X ) ⊗ M is a relative ( D , H ) -Hopf module. roof. Clearly, N ⊗ M is a left D -module. Since N ( X ) is a right H -comodule, N ( X ) ⊗ M also carries a right H -comodule structure for each X ∈ Ob ( D ) . For any f ∈ Hom D ( X, Y ) and n ⊗ m ∈ N ( X ) ⊗ M , we have ρ (( N ⊗ M )( f )( n ⊗ m )) = ρ ( N ( f )( n ) ⊗ m )= ∑ ( N ( f )( n )) ⊗ m ⊗ ( N ( f )( n )) m = ∑ N ( f )( n ) ⊗ m ⊗ f n m = ∑( N ⊗ M )( f )( n ⊗ m ) ⊗ f n m = ∑( N ⊗ M )( f )( n ⊗ m ) ⊗ f ( n ⊗ m ) This shows that N ⊗ M satisfies the condition (5.1) in Definition 5.1.From Lemma 5.2, it follows that the assignment M ↦ N ⊗ M defines a functor N ⊗ ( − ) ∶ Comod - H Ð→ D M H for each N ∈ D M H .From the definition of a co- H -category, it is also clear that the D -module X h ∶ = Hom D ( X, − ) lies in D M H foreach X ∈ Ob ( D ) . Lemma 5.3.
Let M be a relative ( D , H ) -Hopf module and let m ∈ M ( X ) for some X ∈ Ob ( D ) . Then, thereexists a finite dimensional H -comodule W m ⊆ M ( X ) containing m and a morphism η m ∶ X h ⊗ W m Ð→ M in D M H such that η m ( X )( id X ⊗ m ) = m .Proof. Using [7, Theorem 2.1.7], we know that there exists a finite dimensional H -subcomodule W m of M ( X ) containing m . We consider the D -module morphism η m ∶ X h ⊗ W m Ð→ M defined by η m ( Y )( f ⊗ w ) ∶ = M ( f )( w ) for any Y ∈ Ob ( D ) , f ∈ X h ( Y ) and w ∈ W m . We now verify that η m is indeed a morphism in D M H , i.e., η m ( Y ) is H -colinear for each Y ∈ Ob ( D ) : ρ ( η m ( Y )( f ⊗ w )) = ρ ( M ( f )( w )) = ∑ M ( f )( w ) ⊗ f w = ∑ η m ( Y )( f ⊗ w ) ⊗ f w = η m ( Y ) (( f ⊗ w ) ) ⊗ ( f ⊗ w ) Remark 5.4.
It might be tempting to view Lemma 5.3 as a Yoneda correspondence. But, we note that thefinite dimensional H -comodule and the morphism in Lemma 5.3 determined by m ∈ M ( X ) need not be unique.Given a morphism η ∶ M Ð→ N in D M H , it may be easily verified that Ker ( η ) and Coker ( η ) determined bysetting Ker ( η )( X ) ∶ = Ker ( η ( X ) ∶ M ( X ) Ð→ N ( X )) Coker ( η )( X ) ∶ = Coker ( η ( X ) ∶ M ( X ) Ð→ N ( X )) (5.2)for each X ∈ Ob ( D ) are also relative ( D , H ) -Hopf modules. It follows that η ∶ M Ð→ N in D M H is amonomorphism (resp. an epimorphism) if and only if it induces monomorphisms (resp. epimorphisms) η ( X ) ∶ M ( X ) Ð→ N ( X ) of H -comodules for each X ∈ Ob ( D ) . Proposition 5.5.
Let D be a right co- H -category. Then, a module M ∈ D M H is finitely generated as an objectin D - M od if and only if there exists a finite dimensional H -comodule W and an epimorphism (⊕ i ∈ I X i h ) ⊗ W Ð→ M in D M H , for finitely many objects { X i } i ∈ I in D . roof. Let M ∈ D M H be finitely generated as a D -module. Then, there exists a finite collection { m i ∈ el ( M )} i ∈ I such that every y ∈ el ( M ) has the form y = ∑ i ∈ I M ( f i )( m i ) for some f i ∈ Hom D (∣ m i ∣ , ∣ y ∣) . Applying Lemma 5.3,we can obtain for each m i a finite dimensional H -subcomodule W m i ⊆ M (∣ m i ∣) containing m i and a morphism η m i ∶ ∣ m i ∣ h ⊗ W m i Ð→ M in D M H . Setting W ∶ = ⊕ i ∈ I W m i , we have an epimorphism in D M H determined by η ∶ ( ⊕ i ∈ I ∣ m i ∣ h ) ⊗ W Ð→ M , η ( Y )({ f i } i ∈ I ⊗ { w j } j ∈ I ) ∶ = ∑ i ∈ I M ( f i )( w i ) for each Y ∈ Ob ( D ) , f i ∈ ∣ m i ∣ h ( Y ) and w i ∈ W m i .Conversely, let { w , . . . , w k } be a basis of a finite dimensional H -comodule W and { X i } i ∈ I be finitely manyobjects in D such that we have an epimorphism η ∶ ( ⊕ i ∈ I X i h ) ⊗ W Ð→ M in D M H . From the discussion above, it follows that η ( Y ) ∶ ( ⊕ i ∈ I X i h ( Y )) ⊗ W Ð→ M ( Y ) is an epimorphismin Comod - H for each Y ∈ Ob ( D ) . Then, the elements { m ij ∶ = η ( X i )( id X i ⊗ w j )} i ∈ I, ≤ j ≤ k form a family ofgenerators for M as a D -module.We will now show that D M H is a Grothendieck category. This essentially follows from the fact that both D - M od and
Comod - H are Grothendieck categories. We refer the reader, for instance, to [7, Corollary 2.2.8],for a proof of Comod - H being a Grothendieck category. Proposition 5.6.
Let D be a right co- H -category. Then, the category D M H of relative ( D , H ) -Hopf modulesis a Grothendieck category.Proof. Since the categories D - M od and
Comod - H have kernels, cokernels and coproducts (direct sums), so doesthe category D M H . The remaining properties of an abelian category are inherited by D M H from D - M od .Hence, D M H is a cocomplete abelian category. Directs limits are exact in D M H which is also a propertyinherited from D - M od . We are now left to check that D M H has a family of generators. For any M in D M H ,it follows from Lemma 5.3 that we can find an epimorphism ⊕ m ∈ el (M) η m ∶ ⊕ m ∈ el (M) ∣ m ∣ h ⊗ W m Ð→ M in D M H . Thus, the collection { X h ⊗ W } , where X ranges over all objects in D and W ranges over all(isomorphism classes of) finite dimensional H -comodules, forms a generating family for D M H (see, for instance,the proof of [9, Proposition 1.9.1]).For N ∈ D M H , we consider the functor N ⊗ ( − ) ∶ Comod - H Ð→ D M H given by M ↦ N ⊗ M . We see that Comod - H is a Grothendieck category and the functor N ⊗ ( − ) preserves colimits. Therefore, by a classicalresult [14, Proposition 8.3.27(iii)], it has a right adjoint which we denote by R N ∶ D M H Ð→ Comod - H . Wethen define HOM D - Mod ( N , P ) ∶ = R N ( P ) (5.3)for any P ∈ D M H . Thus, we have a natural isomorphism Hom D M H ( N ⊗ M, P ) ≅ Ð→ Hom
Comod - H ( M, HOM D - Mod ( N , P )) (5.4)for N , P ∈ D M H and M ∈ Comod - H . In particular, when D is a right co- H -category with a single object,i.e., a right H -comodule algebra, then N and P are relative Hopf-modules in the classical sense of Takeuchi[26]. Then, using [4, Lemma 2.3], the definition of HOM as in (5.3) recovers the standard definition of rationalmorphisms between relative Hopf modules as in [4, §
2] or [28]. As such, we will refer to
HOM D - Mod ( − , − ) asthe “rational Hom object” in D M H . 21 orollary 5.7. Let N , P ∈ D M H . Then, HOM D - Mod ( N , P ) coH = Hom D M H ( N , P ) .Proof. The result follows by choosing M = k in (5.4) and the fact that Hom
Comod - H ( k, N ) = N coH for any N ∈ Comod - H . Corollary 5.8. If I is an injective in D M H , then HOM D - Mod ( N , I ) is an injective in Comod - H for any N in D M H .Proof. The fact that
HOM D - Mod ( N , − ) ∶ D M H Ð→ Comod - H preserves injectives follows from the fact thatits left adjoint N ⊗ ( − ) ∶ Comod - H Ð→ D 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 M ∈ D M H be a relative ( D , H ) -Hopf module. We consider the functors F = HOM D - Mod ( M , − ) ∶ D M H Ð→ Comod - H N ↦
HOM D - Mod ( M , N ) G = ( − ) coH ∶ Comod - H Ð→ V ect K M ↦ M coH Then, we have the following spectral sequence R p ( − ) coH ( R q HOM D - Mod ( M , − )( N )) ⇒ ( R p + q Hom D M H ( M , − )) ( N ) Proof.
We have ( G ○ F )( N ) = HOM D - Mod ( M , N ) coH = Hom D M H ( M , N ) by Corollary 5.7. By Corollary5.8, the functor F preserves injectives. Since Comod - H has enough injectives, the result now follows fromGrothendieck spectral sequence for composite functors (see [9]).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 ( h ∗ f )( m ) ∶ = ∑ h ∗ ( S − ( m )( f ( m )) ) ( f ( m )) for any h ∗ ∈ H ∗ , f ∈ Hom K ( M, N ) and m ∈ M . We now show that this H ∗ -action can be extended to relative ( D - H ) -Hopf modules. Lemma 5.10.
Let M , N ∈ D M H . Then, Hom D - Mod ( M , N ) is a left H ∗ -module.Proof. For h ∗ ∈ H ∗ and η ∈ Hom D - Mod ( M , N ) , we set ( h ∗ η )( X )( m ) ∶ = ∑ h ∗ ( S − ( m )( η ( X )( m )) )( η ( X )( m )) . (5.5)for all X ∈ Ob ( D ) and m ∈ M ( X ) . We first verify that h ∗ η is indeed an element in Hom D - Mod ( M , N ) . For any f ∈ Hom D ( X, Y ) , we have ( h ∗ η )( Y ) M ( f )( m ) = ∑ h ∗ ( S − (( M ( f )( m )) )( η ( Y )(( M ( f )( m )) )) )( η ( Y )(( M ( f )( m )) )) = ∑ h ∗ ( S − ( f m )( η ( Y )(( M ( f )( m ))) )( η ( Y )(( M ( f )( m ))) ( using (5.1) )= ∑ h ∗ ( S − ( m ) S − ( f )( N ( f )(( η ( X )( m ))) )( N ( f )(( η ( X )( m ))) = ∑ h ∗ ( S − ( m ) S − ( f )( f ) ( η ( X )( m )) ) N (( f ) )( η ( X )( m )) ( using (5.1) )= ∑ h ∗ ( S − ( m ) S − ( f ) f ( η ( X )( m )) ) N ( f )( η ( X )( m )) = N ( f )( h ∗ η )( X )( m ) ( h ∗ g ∗ ) η = h ∗ ( g ∗ η ) and that 1 H ∗ η = η , i.e., εη = η for all h ∗ , g ∗ ∈ H ∗ and η ∈ Hom D - Mod ( M , N ) .The latter equality follows easily and further we see that ( h ∗ ( g ∗ η ))( X )( m ) = ∑ h ∗ ( S − ( m )(( g ∗ η )( X )( m )) )(( g ∗ η )( X )( m )) = ∑ h ∗ ( S − ( m )( g ∗ ( S − (( m ) )( η ( X )(( m ) ))) ( η ( X )(( m ) ))) ) )( g ∗ ( S − (( m ) )( η ( X )(( m ) ))) ( η ( X )(( m ) ))) ) = ∑ h ∗ ( S − ( m )( η ( X )( m )) ) g ∗ ( S − ( m )( η ( X )( m )) )( η ( X )( m )) = ∑( h ∗ g ∗ )( S − ( m )( η ( X )( m )) )( η ( X )( m )) = (( h ∗ g ∗ ) η )( X )( m ) for all X ∈ Ob ( D ) and m ∈ M ( X ) . Lemma 5.11.
Let M , N ∈ D M H and let η ∈ Hom D - Mod ( M , N ) . Then, there is a morphism ρ ( η ) ∈ Hom D - Mod ( M , N ⊗ H ) determined by setting ρ ( η )( X )( m ) ∶ = ∑ ( η ( X )( m )) ⊗ S − ( m )( η ( X )( m )) (5.6) for any X ∈ Ob ( D ) and m ∈ M ( X ) .Proof. Using (5.1) and the fact that η ∈ Hom D - Mod ( M , N ) , we have ρ ( η )( Y )( M ( f )( m )) = ∑ ( η ( Y )( M ( f )( m ))) ⊗ S − ( f m )( η ( Y )( M ( f )( m ))) = ∑ ( N ( f )( η ( X )( m ))) ⊗ S − ( f m )( N ( f )( η ( X )( m ))) = ∑ N ( f )( η ( X )( m )) ⊗ S − ( m ) S − ( f ) f ( η ( X )( m )) = ∑ N ( f )( η ( X )( m )) ⊗ S − ( m )( η ( X )( m )) = ( N ( f ) ⊗ id H ) ρ ( η )( X ) for any f ∈ Hom D ( X, Y ) and m ∈ M ( X ) .We now recall the notion of a rational left H ∗ -module (see, for instance, [7]) which will be used in the next result.Given a left H ∗ -module M , there is a morphism ρ M ∶ M Ð→ Hom K ( H ∗ , M ) corresponding to the canonicalmorphism H ∗ ⊗ M Ð→ M . There is an obvious inclusion M ⊗ H ↪ Hom K ( H ∗ , M ) given by ( m ⊗ h )( h ∗ ) = h ∗ ( h ) m for any m ∈ M , h ∈ H and h ∗ ∈ H ∗ . Definition 5.12. (see [7, Definition 2.2.2] ) A left H ∗ -module M is said to be rational if ρ M ( M ) ⊆ M ⊗ H ,where M ⊗ H is viewed as a subspace of Hom K ( H ∗ , M ) . The full subcategory of rational H ∗ -modules will bedenoted by Rat ( H ∗ - M od ) . If M is a right H -comodule with H -coaction m ↦ ∑ m ⊗ m , then M becomes a left H ∗ -module via the action h ∗ m ∶ = ∑ h ∗ ( m ) m for h ∗ ∈ H ∗ and m ∈ M . This determines a functor Comod - H Ð→ H ∗ - M od
It is well known (see [7, Theorem 2.2.5]) that this functor defines an equivalence of categories between
Comod - H and the subcategory Rat ( H ∗ - M od ) of H ∗ - M od . Proposition 5.13.
Let M , N ∈ D M H and suppose that M is finitely generated as an object in D - M od . Then,
Hom D - Mod ( M , N ) is a right H -comodule. In particular, HOM D - Mod ( M , N ) = Hom D - Mod ( M , N ) . roof. Since M is finitely generated in D - M od , by Proposition 5.5, there exists a finite dimensional H -comodule W and an epimorphism η ∶ (⊕ i ∈ I X i h ) ⊗ W Ð→ M in D M H , for finitely many objects { X i } i ∈ I in D . From the description of epimorphisms in D M H in (5.2), weknow that η is also an epimorphism in D - M od . The map
Hom ( η, N ) ∶ Hom D - Mod ( M , N ) ↪ ⊕ i ∈ I Hom D - Mod ( X i h ⊗ W, N ) is therefore a monomorphism for each N ∈ D M H . Using the fact that η ( Y ) is H -colinear for each Y ∈ Ob ( D ) ,we will now verify that the morphism Hom ( η, N ) is H ∗ -linear. For any h ∗ ∈ H ∗ , ξ ∈ Hom D - Mod ( M , N ) , Y ∈ Ob ( D ) and ˜ f ⊗ w ∈ ( ⊕ i ∈ I X i h ( Y )) ⊗ W , we have ( Hom ( η, N )( h ∗ ξ )) ( Y )( ˜ f ⊗ w )= (( h ∗ ξ ) ○ η )( Y )( ˜ f ⊗ w ) = ( h ∗ ξ )( Y ) ( η ( Y )( ˜ f ⊗ w ))= ∑ h ∗ ( S − (( η ( Y )( ˜ f ⊗ w )) ) ( ξ ( Y )(( η ( Y )( ˜ f ⊗ w )) ) )( ξ ( Y )(( η ( Y )( ˜ f ⊗ w )) ) = ∑ h ∗ ( S − (( ˜ f ⊗ w ) ) ( ξ ( Y ) ( η ( Y )( ˜ f ⊗ w ) ) ) )( ξ ( Y ) ( η ( Y )( ˜ f ⊗ w ) ) ) = ∑ h ∗ ( S − (( ˜ f ⊗ w ) ) (( ξ ○ η )( Y )( ˜ f ⊗ w ) )) )(( ξ ○ η )( Y )( ˜ f ⊗ w ) )) = ( h ∗ ( ξ ○ η )) ( Y )( ˜ f ⊗ w )= ( h ∗ Hom ( η, N )( ξ )) ( Y )( ˜ f ⊗ w ) This shows that
Hom D - Mod ( M , N ) is an H ∗ -submodule of ⊕ i ∈ I Hom D - Mod ( X i h ⊗ W, N ) .For each i ∈ I , we now prove that ρ ∶ Hom D - Mod ( X i h ⊗ W, N ) Ð→ Hom D - Mod ( X i h ⊗ W, N ⊗ H ) , as defined in(5.6), gives an H -comodule structure on Hom D - Mod ( X i h ⊗ W, N ) . Since W is finite dimensional, we have Hom D - Mod ( X i h ⊗ W, N ⊗ H ) ≅ Hom K ( W, Hom D - Mod ( X i h , N ⊗ H ))≅ Hom K ( W, N ( X i ) ⊗ H ) ≅ Hom K ( W, N ( X i )) ⊗ H ≅ Hom D - Mod ( X i h ⊗ W, N ) ⊗ H This gives a well defined morphism ρ ∶ Hom D - Mod ( X i h ⊗ W, N ) Ð→ Hom D - Mod ( X i h ⊗ W, N ⊗ H ) ≅ Hom D - Mod ( X i h ⊗ W, N ) ⊗ H (5.7)We will verify that (5.7) gives a right H -coaction. For this, we need to show that for any ζ ∈ Hom D - Mod ( X i h ⊗ W, N ) , we have ( ρ ⊗ id ) ρ ( ζ ) = ( id ⊗ ∆ ) ρ ( ζ ) and ( id ⊗ ε ) ρ ( ζ ) = ζ . The latter equality is easy to verify. By (5.7),we know that ρ ( ζ ) = ∑ ζ ⊗ ζ ∈ Hom D - Mod ( X i h ⊗ W, N ) ⊗ H . Thus, for any X ∈ Ob ( D ) and u ∈ X i h ( X ) ⊗ W ,we have (( ρ ⊗ id ) ρ ( ζ )) ( X )( u ) = ∑ ρ ( ζ )( X )( u ) ⊗ ζ = ∑( ζ ( X )( u )) ⊗ S − ( u )( ζ ( X )( u )) ⊗ ζ = ∑( ζ ( X )( u )) ⊗ S − ( u )( ζ ( X )( u )) ⊗ S − ( u )( ζ ( X )( u )) = ∑ ζ ( X )( u ) ⊗ ζ ⊗ ζ . The third equality above follows by applying ρ N ( X ) ⊗ id H on the equality ∑ ζ ( X )( u ) ⊗ ζ = ρ ( ζ )( X )( u ) andthe last one is obtained by applying id H ⊗ ∆ on ∑ ζ ( X )( u ) ⊗ ζ = ∑ ( ζ ( X )( u )) ⊗ S − ( u )( ζ ( X )( u )) . Thus,we have shown that Hom D - Mod ( X i h ⊗ W, N ) is a right H -comodule.Moreover, the H ∗ -action on Hom D - Mod ( X i h ⊗ W, N ) as in (5.5) is given precisely by the H -coaction as in(5.6). Therefore, Hom D - Mod ( X i h ⊗ W, N ) is a rational H ∗ -module. Since the category of rational H ∗ -modules24ontains direct sums (it is equivalent to Comod - H ), it follows that ⊕ i ∈ I Hom D - Mod ( X i h ⊗ W, N ) is also a rational H ∗ -module. Being an H ∗ -submodule of ⊕ i ∈ I Hom D - Mod ( X i h ⊗ W, N ) , it is now clear that Hom D - Mod ( M , N ) is also a rational left H ∗ -module and hence a right H -comodule.It may be verified that the functor Hom D - Mod ( M , − ) ∶ D M H Ð→ Comod - H is right adjoint to the functor M ⊗− ∶ Comod - H Ð→ D M H given by N ↦ M ⊗ N . Thus, by the uniqueness of adjoints, we have Hom D - Mod ( M , − ) = HOM D - Mod ( M , − ) .A morphism N Ð→ N ′ in D M H induces a morphism of functors N ⊗ ( − ) Ð→ N ′ ⊗ ( − ) and hence a morphism R N ′ Ð→ R N of their respective right adjoints. Thus, for any L ∈ D M H , we have a functor HOM D - Mod ( − , L ) ∶ ( D M H ) op Ð→ Comod - H which takes N to HOM D - Mod ( N , L ) = R N ( L ) . Proposition 5.14. (1) For any L ∈ D M H , the functor HOM D - Mod ( − , L ) ∶ ( D M H ) op Ð→ Comod - H is leftexact, i.e., it preserves kernels.(2) If I is injective in D M H , then HOM D - Mod ( − , I ) is exact.(3) If I is injective in D M H , then HOM D - Mod ( − , I ) takes every short exact sequence in D M H to a split shortexact sequence in Comod - H .Proof. (1) Let η ∶ M Ð→ N be a morphism in D M H and let P ∶ = Coker ( η ) . Then, for any T ∈ Comod - H , Coker ( η ⊗ id T ∶ M ⊗ T Ð→ N ⊗ T ) = P ⊗ T . From the adjunction in (5.4), we now have Hom
Comod - H ( T, HOM D - Mod ( P , L )) ≅ Hom D M H ( P ⊗ T, L )≅ Ker ( Hom D M H ( N ⊗ T, L ) Ð→ Hom D M H ( M ⊗ T, L ))≅ Ker ( Hom
Comod - H ( T, HOM D - Mod ( N , L )) Ð→ Hom
Comod - H ( T, HOM D - Mod ( M , L )))≅ Hom
Comod - H ( T, Ker ( HOM D - Mod ( N , L ) Ð→ HOM D - Mod ( M , L ))) for any T ∈ Comod - H . From Yoneda Lemma, it follows that HOM D - Mod ( P , L ) = Ker ( HOM D - Mod ( N , L ) Ð→ HOM D - Mod ( M , L )) (2) Let 0 Ð→ M Ð→ N Ð→ P Ð→ D M H . From (1), we already know that0 Ð→ HOM D - Mod ( P , I ) Ð→ HOM D - Mod ( N , I ) q Ð→ HOM D - Mod ( M , I ) (5.8)is exact. We need to show that q is an epimorphism. For any T ∈ Comod - H , we notice that 0 Ð→ M ⊗ T Ð→N ⊗ T Ð→ P ⊗ T Ð→ D M H . If I is an injective object in D M H , we see that0 Ð→ Hom D M H ( P ⊗ T, I ) Ð→ Hom D M H ( N ⊗ T, I ) Ð→ Hom D M H ( M ⊗ T, I ) Ð→ K -vector spaces. Using the adjunction in (5.4), it follows that0 ÐÐÐÐ→
Hom
Comod - H ( T, HOM D - Mod ( P , I )) ÐÐÐÐ→
Hom
Comod - H ( T, HOM D - Mod ( N , I )) Hom ( T,q ) ÐÐÐÐÐÐ→
Hom
Comod - H ( T, HOM D - Mod ( M , I )) ÐÐÐÐ→
V ect K . By setting T = HOM D - Mod ( M , I ) in (5.9), we see that there exists a morphism f ∶ HOM D - Mod ( M , I ) Ð→ HOM D - Mod ( N , I ) of H -comodules such that q ○ f is the identity on HOM D - Mod ( M , I ) .This shows that q ∶ HOM D - Mod ( N , I ) Ð→ HOM D - Mod ( M , I ) is an epimorphism. The result of (3) is clearfrom the proof of (2). 25 roposition 5.15. Let D be a left noetherian right co- H -category and let M ∈ D M H be finitely generated asan object in D - M od . If I is an injective object in D M H , then Ext p D - Mod ( M , I ) = for all p > .Proof. Since M ∈ D M H is finitely generated in D - M od , by Proposition 5.5, there exists a finite dimensional H -comodule W and an epimorphism η ∶ P ∶ = ( n ⊕ i = X i h ) ⊗ W Ð→ M in D M H for finitely many objects { X i } ≤ i ≤ n in D . Then, K ∶ = Ker ( η ) is a subobject of P in D M H . Since D isleft noetherian, P is a noetherian left D -module (see, for instance, [20, § K = Ker ( η ) of P is finitely generated as an object in D - M od . Therefore, we obtain a finite dimensional H -comodule W and an epimorphism η ∶ P ∶ = ( n ⊕ j = Y j h ) ⊗ W Ð→ K in D M H for finitely many objects { Y j } ≤ j ≤ n in D . Since W and W are finite dimensional K -vector spaces,clearly P and P are free left D -modules. Moreover, Im ( η ) = K = Ker ( η ) . Continuing in this way, we canconstruct a free resolution of the module M in the category D - M od : P ∗ = ⋯ Ð→ P i Ð→ ⋯ Ð→ P Ð→ P Ð→ M Ð→ p D - Mod ( M , I ) = H p ( Hom D - Mod ( P ∗ , I )) , ∀ p > M and { P i } i ≥ are finitely generated in D - M od , it follows from Proposition 5.13 that
HOM D - Mod ( M , I ) = Hom D - Mod ( M , I ) and HOM D - Mod ( P i , I ) = Hom D - Mod ( P i , I ) . From Proposition 5.14, we know that thefunctor HOM D - Mod ( − , I ) is exact and it follows that Ext p D - Mod ( M , I ) = H p ( HOM D - Mod ( P ∗ , I )) = p > Proposition 5.16.
Let D be a left noetherian right co- H -category. Let M , N ∈ D M H with M finitely generatedas an object in D - M od . If E ∗ is an injective resolution of N in D M H , then Ext p D - Mod ( M , N ) = R p HOM D - Mod ( M , N ) = H p ( Hom D - Mod ( M , E ∗ )) , ∀ p ≥ . Proof.
Let P ∗ be the free resolution of M in D - M od constructed as in the proof of Proposition 5.15. Then, wehave Ext p D - Mod ( M , N ) = H p ( Hom D - Mod ( P ∗ , N )) = H p ( HOM D - Mod ( P ∗ , N )) where the second equality follows from Proposition 5.13. Since HOM D - Mod ( P ∗ , N ) is a complex in Comod - H and Comod - H is an abelian category, it follows that H p ( HOM D - Mod ( P ∗ , N )) ∈ Comod - H . Hence, we mayconsider the family { Ext p D - Mod ( M , − )} p ≥ as a δ − functor from D M H to Comod - H .By Proposition 5.15, Ext p D - Mod ( M , I ) = p > I ∈ D M H . Since D M H has enoughinjectives, it follows that each Ext p D - Mod ( M , − ) ∶ D M H Ð→ Comod - H is effaceable (see, for instance, [12, § III.1]).Since D M H has enough injectives, we can consider the right derived functors R p HOM D - Mod ( M , − ) ∶ D M H Ð→ Comod - H p ≥ p =
0, we notice that Ext D - Mod ( M , − ) = Hom D - Mod ( M , − ) = HOM D - Mod ( M , − ) = R HOM D - Mod ( M , − ) as functors from D M H to Comod - H . Since each Ext p D - Mod ( M , − ) is effaceable for p >
0, we see that the family { Ext p D - Mod ( M , − )} p ≥ forms a universal δ -functor and it follows from [12, Corollary III.1.4] thatExt p D - Mod ( M , − ) = R p HOM D - Mod ( M , − ) ∶ D M H Ð→ Comod - H p ≥
0. Therefore, we haveExt p D - Mod ( M , N ) = ( R p HOM D - Mod ( M , − )) ( N ) = H p ( HOM D - Mod ( M , E ∗ )) = H p ( Hom D - Mod ( M , E ∗ )) Recall that by Proposition 5.13, for any M ∈ D M H with M finitely generated as an object in D - M od , we have
Hom D - Mod ( M , N ) = HOM D - Mod ( M , N ) ∈ Comod - H . Theorem 5.17.
Let D be a left noetherian right co- H -category. Let M ∈ D M H with M finitely generated asan object in D - M od . We consider the functors F = Hom D - Mod ( M , − ) ∶ D M H Ð→ Comod - H N ↦
Hom D - Mod ( M , N ) G = ( − ) coH ∶ Comod - H Ð→ V ect K M ↦ M coH Then, we have the following spectral sequence R p ( − ) coH ( Ext q D - Mod ( M , N )) ⇒ ( R p + q Hom D M H ( M , − )) ( N ) Proof.
By Corollary 5.7, we have ( G ○ F )( N ) = Hom D - Mod ( M , N ) coH = HOM D - Mod ( M , N ) coH = Hom D M H ( M , N ) By definition, R q F ( N ) = H q ( F ( E ∗ )) = H q ( Hom D - Mod ( M , E ∗ )) (5.10)where { E ∗ } is an injective resolution of N in D M H . Applying Corollary 5.16, we obtain Ext q D - Mod ( M , N ) = R q F ( N ) . For any injective object I in D M H , we know that F ( I ) = Hom D - Mod ( M , I ) = HOM D - Mod ( M , I ) is injective in Comod - 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 [9]). References [1] A. Banerjee,
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