Frobenius complexes and the homotopy colimit of a diagram of posets over a poset

TL;DR

This paper investigates the homotopy types of Frobenius complexes associated with affine monoids, especially when constructed by gluing simpler monoids, and explores implications for the Poincaré series of their algebra's torsion groups.

Contribution

It expresses the homotopy types of Frobenius complexes of glued affine monoids in terms of simpler components, advancing understanding of their topological and algebraic properties.

Findings

Homotopy types of Frobenius complexes can be decomposed for glued affine monoids.

The approach relates Frobenius complexes to the Poincaré series of monoid algebra torsion groups.

Provides a method to analyze the topology of affine monoids via their building blocks.

Abstract

An affine monoid is an additive monoid which is cancellative, pointed and finitely generated. An affine monoid $\Lambda$ has the partial order defined by $\lambda \le \lambda + \mu$. The Frobenius complex is the order complex of an open interval of $\Lambda$ with respect to this partial order. The reduced homology of the Frobenius complex is related to the torsion group of the monoid algebra $K[\Lambda]$. In this paper, we pay attention to homotopy types of Frobenius complexes, and we express the homotopy types of the Frobenius complexes of $\Lambda$ in terms of those of $\Lambda_1$ and $\Lambda_2$ when $\Lambda$ is an affine monoid obtained by gluing two affine monoids $\Lambda_1$ and $\Lambda_2$ with one relation. We also state an application to the Poincar\'e series of the torsion group of the monoid algebra.