Extension category algebras and LHS--spectral sequences

2025/07/28 by Wu, Mawei
#FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2507.20588

Abstract

Let C be a small category, \mathfrakA be a precosheaf of unital k-algebras on C and \mathfrakM be an \mathfrakA-bimodule. We introduce two new notions, namely, the Grothendieck construction Gr_C(\mathfrakA, \mathfrakM) of \mathfrakA and \mathfrakM, as well as the extension category algebra \mathfrakA \ltimes \mathfrakM. The extension category algebra contains the trivial extension algebra and the skew category algebra as special cases. If C is object-finite, we prove that the category of modules of Gr_C(\mathfrakA, \mathfrakM) is equivalent to the category of modules over \mathfrakA \ltimes \mathfrakM. Finally, we obtain two LHS-spectral sequences about Gr_C(\mathfrakA, \mathfrakN) for a right \mathfrakA-module \mathfrakN.

Citations

• Thomason cohomology and Quillen's Theorem A
• Pseudoskew category algebras and modules over representations of small categories
• Skew category algebras, twisted tensor product algebras and induction of precosheaves
• LHS-spectral sequences for regular extensions of categories
• The category of modules on an n-trivial extension: the basic properties
• Homological dimension formulas for trivial extension algebras
• On n-Trivial Extensions of Rings
• Hochschild cohomology of triangular matrix algebras

Related

• LHS-spectral sequences for regular extensions of categories
• Non-abelianess of the category of modules over a sum-id bipresheaf of rings
• Pseudoskew category algebras and modules over representations of small categories
• Ring extension problem, Shukla cohomology and Ann-category theory
• Abelian extensions of algebras in congruence-modular varieties