The Andr\'e-Quillen cohomology of commutative monoids

TL;DR

This paper establishes a connection between Quillen cohomology of commutative monoids and Andre9-Quillen cohomology of graded commutative rings, expanding the theoretical framework and providing new computational tools.

Contribution

It shows that Beck modules for commutative monoids are modules over a graded ring and relates Quillen cohomology to Andre9-Quillen cohomology, generalizing previous results.

Findings

Beck modules for commutative monoids are modules over a graded commutative ring.

Quillen cohomology of commutative monoids is a special case of Andre9-Quillen cohomology for graded rings.

Development of grading formalism and modification of Harrison cochain complex for cohomology computation.

Abstract

We observe that Beck modules for a commutative monoid are exactly modules over a graded commutative ring associated to the monoid. Under this identification, the Quillen cohomology of commutative monoids is a special case of Andr\'e-Quillen cohomology for graded commutative rings, generalizing a result of Kurdiani and Pirashvili. To verify this we develop the necessary grading formalism. The partial cochain complex developed by Pierre Grillet for computing Quillen cohomology appears as the start of a modification of the Harrison cochain complex suggested by Michael Barr.