Semigroup cohomology as a derived functor

TL;DR

This paper develops a new extension of the category of 0-modules, enabling 0-cohomology to be viewed as a derived functor, and applies this to determine the cohomological dimension of 0-free monoids.

Contribution

It introduces an extension of the category of 0-modules that allows 0-cohomology to be interpreted as a derived functor, advancing the theoretical framework.

Findings

Calculated the cohomological dimension of 0-free monoids.

Established the 0-cohomology functor as a derived functor in the extended category.

Extended the category of 0-modules analogously to prior work on small categories.

Abstract

In this work we construct an extension for the category of 0-modules by analogy with [H.-J. Baues and G. Wirshing, Cohomology of small categories, J. Pure Appl. Algebra, 38(1985), 187-211]. The 0-cohomology functor becomes a derived functor in the extended category. As an application of this construction we calculate the cohomological dimension of so-called 0-free monoids.