Categories of modules, comodules and contramodules over representations

We study and relate categories of modules, comodules and contramodules over a representation of a small category taking values in (co)algebras, in a manner similar to modules over a ringed space. As a result, we obtain a categorical framework which incorporates all the adjoint functors between these categories in a natural manner. Various classical properties of coalgebras and their morphisms arise naturally within this theory. We also consider cartesian objects in each of these categories, which may be viewed as counterparts of quasi-coherent sheaves over a scheme. We study their categorical properties using cardinality arguments. Our focus is on generators for these categories and on Grothendieck categories, because the latter may be treated as replacements for noncommutative spaces.Comment: Several update

Comodule theories in Grothendieck categories and relative Hopf objects

We develop the categorical algebra of the noncommutative base change of a comodule category by means of a Grothendieck category $\mathfrak S$. We describe when the resulting category of comodules is locally finitely generated, locally noetherian or may be recovered as a coreflective subcategory of the noncommutative base change of a module category. We also introduce the category ${_A}\mathfrak S^H$ of relative $(A,H)$-Hopf modules in $\mathfrak S$, where $H$ is a Hopf algebra and $A$ is a right $H$-comodule algebra. We study the cohomological theory in ${_A}\mathfrak S^H$ by means of spectral sequences. Using coinduction functors and functors of coinvariants, we study torsion theories and how they relate to injective resolutions in ${_A}\mathfrak S^H$. Finally, we use the theory of associated primes and support in noncommutative base change of module categories to give direct sum decompositions of minimal injective resolutions in ${_A}\mathfrak S^H$.Comment: Minor update