Homotopy type of Frobenius complexes

TL;DR

This paper determines the homotopy type of Frobenius complexes for specific submonoids of N^d and applies these results to compute the rational multigraded Poincare series of certain quotient algebras.

Contribution

It explicitly characterizes the homotopy type of Frobenius complexes for particular submonoids and derives the rational form of the Poincare series for related quotient algebras.

Findings

Homotopy type of Frobenius complex for submonoids generated by two coprime integers

Homotopy type for submonoids of N^2 generated by three linearly independent elements

Multigraded Poincare series of K[x,y,z]/(x^p y^q - z^r) is rational

Abstract

A submonoid A of N^d has a natural order defined by a <= a + b for elements a and b of A. The Frobenius complex is the order complex of an open interval of A with respect to this order. In this paper, the homotopy type of the Frobenius complex of A is determined when A is the submonoid of N generated by two relatively prime integers, or the submonoid of N^2 generated by three elements of which any two are linearly independent. As an application, the multigraded Poincare series of the quotient algebra K[x, y, z] / (x^p y^q - z^r) over a field K is determined and proved to be rational.