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Cohomology of Modules Over
$H$-categories and Co-
$H$-categories
- Part of
-
- Journal:
- Canadian Journal of Mathematics / Volume 72 / Issue 5 / October 2020
- Published online by Cambridge University Press:
- 06 August 2019, pp. 1352-1385
- Print publication:
- October 2020
-
- Article
- Export citation
-
Let
$H$ be a Hopf algebra. We consider 
$H$-equivariant modules over a Hopf module category 
${\mathcal{C}}$ as modules over the smash extension 
${\mathcal{C}}\#H$. We construct Grothendieck spectral sequences for the cohomologies as well as the 
$H$-locally finite cohomologies of these objects. We also introduce relative 
$({\mathcal{D}},H)$-Hopf modules over a Hopf comodule category 
${\mathcal{D}}$. 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 
$\text{Hom}$ objects and higher derived functors of coinvariants.
