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

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 Poincaré series of the torsion group of the monoid algebra.