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Cohomology of Modules Over $H$-categories and Co-$H$-categories

Part of: Hopf algebras, quantum groups and related topics Rings and algebras arising under various constructions Abelian categories

Published online by Cambridge University Press:  06 August 2019

Mamta Balodi ,
Abhishek Banerjee  and
Samarpita Ray
Mamta Balodi
Affiliation:
Department of Mathematics, Indian Institute of Science, Bangalore560012, India Email: mamta.balodi@gmail.comabhishekbanerjee1313@gmail.comray.samarpita31@gmail.com
Abhishek Banerjee
Affiliation:
Department of Mathematics, Indian Institute of Science, Bangalore560012, India Email: mamta.balodi@gmail.comabhishekbanerjee1313@gmail.comray.samarpita31@gmail.com
Samarpita Ray
Affiliation:
Department of Mathematics, Indian Institute of Science, Bangalore560012, India Email: mamta.balodi@gmail.comabhishekbanerjee1313@gmail.comray.samarpita31@gmail.com
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Abstract

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.

Keywords

H-categoryco-H-categoryH-equivariant modulerelative Hopf module

MSC classification

Primary: 16S40: Smash products of general Hopf actions
Secondary: 16T05: Hopf algebras and their applications 18E05: Preadditive, additive categories
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 5 , October 2020 , pp. 1352 - 1385
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Copyright
© Canadian Mathematical Society 2019

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Footnotes

Author M.B. was supported by SERB Fellowship PDF/2017/000229. Author A.B. was partially supported by SERB Matrics fellowship MTR/2017/000112.

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