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 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 ASH of relative (A, H)-Hopf modules in S, where H is a Hopf algebra and A is a right H-comodule algebra. We study the cohomological theory in ASH by means of spectral sequences. Using coinduction functors and functors of coinvariants, we study torsion theories and how they relate to injective resolutions in ASH. 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 ASH. (c) 2024 Elsevier B.V. All rights reserved.