Cohomology of comodules (original) (raw)

A Construction of Injective Cogenerators

Journal of the London Mathematical Society, 1990

Throughout the paper, we assume that K is an algebraically closed field, and we always use the term 'module' to mean left module. With terminology suggested by [1], we say that a A^-algebra A has a multiplicative basis B, if B is a basis of the underlying K-vector space of A, and the product of two elements of B is either an element of B or zero. If A is a A>algebra, then we denote by ,4-mod (respectively mod-y4) the category of all left modules A M (respectively right modules M A). For any A>vector space K V, we assume that the abelian group H o m^^.^K) has its usual structure of left ^-module [3, Proposition 3.5]. Hence, i f / e H o m^G^^F) and aeA, then the element afe¥iom K (K A A , K V) is defined by the formula (af)(a') =Aa'a) for any a'eA. In particular, for brevity, we denote by A* the left A-module Hom K (K A A , K K). In Section 1, we show that, if A is a A^-algebra with a multiplicative basis, then we may describe A* in a concrete way, by means of a quiver. In Section 2, we use this result to obtain some information on the size of A* and of its socle. Let where {Q(i):iel} is a complete set of (representatives of the isomorphism classes of) indecomposable injective ^4-modules with simple socles. Then clearly Q is an injective cogenerator and so is A*. In Section 2 we use our description of A* to show that while in certain cases these two injective cogenerators are related, nevertheless, in general, they do not determine each other. To see this, it suffices to consider two examples. First of all, let A be the free algebra K[x]. In this case, A* is isomorphic to a summand of Q, and A* is isomorphic to Q if and only if A* and Q have the same cardinality. Next, let A be the free algebra K(x,y} in two non-commutative variables x and y. In this case, the socle of A* is extremely large, and A* cannot be embedded in Q n for any n^\. Indeed, for any n ^ 1, there is a simple module S, with dimension n over K, such that dim K Hom^S 1 , A*) = n. Moreover, there is a simple module S, with endomorphism ring K, such that dim K Hom i4 (5 r , A*) is as large as might be expected; that is, dim K \iom A (S,A*) is equal to the cardinality of A*. I would like to thank Professor C. M. Ringel for his many useful suggestions during the Ottawa-Moosonee Workshop in Algebra (August 1987), where some of the results contained in this note were announced. 1 With all the preceding notation, we introduce some conventions and definitions used throughout the paper.

R A ] 2 9 Ju l 2 01 9 Cohomology of modules over H-categories and coH-categories

2019

Let H be a Hopf algebra. We consider H-equivariant modules over a Hopf module category C as modules over the smash extension 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 (D,H)-Hopf modules over a Hopf 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 objects and higher derived functors of coinvariants. MSC(2010) Subject Classification: 16S40, 16T05, 18E05

Relative injectivity and CS-modules

International Journal of Mathematics and Mathematical Sciences, 1994

In this paper we show that a direct decomposition of modules M N, with N homologically independent to the inJective hull of H, is a CS-module if and only if N is injective relative to H and both of M and N are CS-modules. As an application, we prove that a direct sum of a non-singular semisimple module and a quasi-continuous module with zero socle is quasi-continuous. This result is known for quasi-inOective modules. But when we confine ourselves to CS-modules we need no conditions on their socles. Then we investigate direct sums of CS-modules which are pairwise relatively inective. We show that every flnite direct sum of such modules is a CS-module. This result is known for quasi-continuous modules. For the case of infinite direct sums, one has to add an extra condition. Finally, we briefly discuss modules in which every two direct summands are relatively inective.

On Galois Comodules

Communications in Algebra, 2006

Generalising the notion of Galois corings, Galois comodules were introduced as comodules P over an A-coring C for which PA is finitely generated and projective and the evaluation map µC : Hom C (P, C) ⊗S P → C is an isomorphism (of corings) where S = End C (P ). It was observed that for such comodules the functors HomA(P, −) ⊗S P and − ⊗A C from the category of right A-modules to the category of right C-comodules are isomorphic. In this note we call modules P with this property Galois comodules without requiring PA to be finitely generated and projective. This generalises the old notion with this name but we show that essential properties and relationships are maintained. These comodules are close to being generators and have some common properties with tilting (co)modules. Some of our results also apply to generalised Hopf Galois (coalgebra Galois) extensions. 1 in [4, 18.26] that this condition implies that the functors Hom A (P, −) ⊗ S P and − ⊗ A C from the right A-modules to the right C-comodules are isomorphic.

Supports in abstract module categories, local cohomology objects and spectral sequences

2022

We work with a strongly locally noetherian Grothendieck category S and we consider the category SR of R-module objects in S introduced by Popescu, where R is a commutative and noetherian k-algebra. Then, SR may be seen as an abstract category of modules over a noncommutative base change of R. Using what we call R-elementary objects in SR and their injective hulls, we develop a theory of supports and associated primes in the abstract module category SR. We use these methods to study associated primes of local cohomology objects in SR. In fact, we use a more general framework, extending the local cohomology with respect to a pair of ideals I, J ⊆ R introduced by Takahashi, Yoshino and Yoshizawa. We give a finiteness condition for the set of associated primes of local cohomology objects. Thereafter, we apply our theory to study a general functorial setup that requires certain conditions on the injective hulls of R-elementary objects and gives us spectral sequences for derived functors associated to two variable local cohomology objects, as well as generalized local cohomology and also generalized Nagata ideal transforms on SR. We note that this framework will give new results, even in the case we take S to be the category of modules over a noncommutative algebra that is strongly locally noetherian.