An A(oo)-structure on the cohomology ring of the symmetric group Sp with coefficients in Fp

TL;DR

This paper constructs an A(oo)-structure on the cohomology ring of the symmetric group over a finite field, providing a minimal model for the associated dg-algebra, advancing understanding of algebraic structures in group cohomology.

Contribution

It introduces an A(oo)-structure on the Ext algebra of the symmetric group over a finite field, offering a new minimal model for the dg-algebra of endomorphisms.

Findings

Constructed a projective resolution of the trivial module

Equipped the Ext algebra with an A(oo)-structure

Provided a minimal model for the dg-algebra of endomorphisms

Abstract

Let p be a prime. Let FpSp be the group algebra of the symmetric group over the finite field Fp with |Fp|=p$. Let Fp be the trivial FpSp-module. We present a projective resolution PRes Fp of the module Fp and equip the Yoneda algebra Ext(Fp,Fp) with an A(oo)-structure such that Ext(Fp,Fp) becomes a minimal model of the dg-algebra Hom(PRes Fp, PRes Fp).