Projective Modules and Cohomology for Integral Basic Algebras

TL;DR

This paper investigates how certain basic algebras over different characteristics exhibit similar cohomological properties and Cartan numbers when specific conditions are met, extending understanding of algebra behavior across characteristics.

Contribution

It introduces conditions under which the cohomology groups and graded Cartan numbers of basic algebras are equal across characteristics, including examples like Solomon descent algebras and nilCoxeter algebras.

Findings

Equalities of cohomology group dimensions between simple modules

Equalities of graded Cartan numbers under certain hypotheses

Examples include Solomon descent algebras and nilCoxeter algebra

Abstract

Algebras defined over fields of characteristic zero and positive characteristic usually do not behave the same way. However, for certain algebras, for example the group algebras, they behave the same way as the characteristic zero case at "good enough" prime. In this paper, we initiate the study of this topic by imposing increasingly strong hypotheses on basic algebras. When the algebras satisfy the right hypotheses, we have equalities of the dimensions of their cohomology groups between simple modules and equalities of graded Cartan numbers. The examples include the Solomon descent algebras of finite Coxeter groups at large enough primes, nilCoxeter algebra, and certain finite semigroup algebras at an arbitrary prime.