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 ${}_A{\mathfrak{S}}^H$.