Sheaves on finite categories and their applications

Title

Sheaves on finite categories and their applications

Speaker

Fei XU (Shantou University)

Time and Location

Dec. 06, 2022, Tuesday, 18:30-19:30, E508 (Education Building, 5th floor)

Abstract

In representations and cohomology of finite groups, one finds many finite categories,  such as the orbit categories and the fusion systems. A finite category C, along with a Grothendieck topology J, is called a finite site, written as C=(C,J). Sheaves on finite sites may be used to construct group representations and to compute important invariants in modular representation theory. It motivates us to investigate modules on ringed finite sites. Let  k be a commutative ring with identity and R be a sheaf of k-algebras on C.  We prove that the module category of R is equivalent to that of an associative algebra R[D], the skew category algebra of R over an idempotent complete strictly full subcategory D of C.

As an application,  we can identify the two cohomological formulas for  computing  the group of endotrivial modules, by P. Balmer and J. Grodal. We may also establish an isomorphism between the Hochschild cohomology of a presheaf of k-algebras, introduced by Gerstenhaber and Schack, and the Hochschild cohomology of some skew category algebra.

Biography

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