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We develop the categorical algebra of the noncommutative base change of acomodule category by means of a Grothendieck category $\mathfrak S$. Wedescribe when the resulting category of comodules is locally finitelygenerated, locally noetherian or may be recovered as a coreflective subcategoryof the noncommutative base change of a module category. We also introduce thecategory ${_A}\mathfrak S^H$ of relative $(A,H)$-Hopf modules in $\mathfrak S$,where $H$ is a Hopf algebra and $A$ is a right $H$-comodule algebra. We studythe cohomological theory in ${_A}\mathfrak S^H$ by means of spectral sequences.Using coinduction functors and functors of coinvariants, we study torsiontheories and how they relate to injective resolutions in ${_A}\mathfrak S^H$.Finally, we use the theory of associated primes and support in noncommutativebase change of module categories to give direct sum decompositions of minimalinjective resolutions in ${_A}\mathfrak S^H$.

Comodule theories in Grothendieck categories and relative Hopf objects