Homological Algebra for Commutative Monoids

TL;DR

This paper develops a homological algebra framework for commutative monoids, exploring their structure, modules, K-theory, extensions, and homotopy theory, drawing parallels with commutative ring theory.

Contribution

It introduces homological algebra concepts to commutative monoids, including projective modules, K-theory, extensions, and model structures, extending classical algebraic tools to this setting.

Findings

Classification of projective A-sets by rank

Calculation of K_0 and K_1 groups for monoids

Description of extensions and homotopy structures for A-sets

Abstract

We first study commutative, pointed monoids providing basic definitions and results in a manner similar commutative ring theory. Included are results on chain conditions, primary decomposition as well as normalization for a special class of monoids which lead to a study monoid schemes, divisors, Picard groups and class groups. It is shown that the normalization of a monoid need not be a monoid, but possibly a monoid scheme. After giving the definition of, and basic results for, $A$-sets, we classify projective $A$-sets and show they are completely determine by their rank. Subsequently, for a monoid $A$, we compute $K_0$ and $K_1$ and prove the Devissage Theorem for $G_0$. With the definition of short exact sequence for $A$-sets in hand, we describe the set $Ext(X,Y)$ of extensions for $A$-sets $X,Y$ and classify the set of square-zero extensions of a monoid $A$ by an $A$-set $X$ using the Hochschild cosimplicial set. We also examine the projective model structure on simplicial $A$-sets showcasing the difficulties involved in computing homotopy groups as well as determining the derived category for a monoid. The author defines the category $\operatorname{Da}(\mathcal{C})$ of double-arrow complexes for a class of non-abelian categories $\mathcal{C}$ and, in the case of $A$-sets, shows an adjunction with the category of simplicial $A$-sets.