The Survival Complex

TL;DR

This paper introduces the 'survival complex' associated with semigroups, explores its properties in Artinian monomial rings, and connects the complex's topology to algebraic features like Gorenstein conditions and ring decompositions.

Contribution

It defines the survival complex for semigroups, characterizes its isolated points in monomial rings, and establishes links between the complex's structure and algebraic properties such as Gorensteinness and ring factorizations.

Findings

Survival complex always has an isolated point in Artinian monomial rings.

Pure power monomial ideals correspond to rings with a unique isolated point.

The complex's connected components relate to the ring's fiber product factors.

Abstract

We introduce a new way to associate a simplicial complex called the \emph{survival complex} to a commutative semigroup with zero. Restricting our attention to the semigroup of monomials arising from an Artinian monomial ring, we determine that any such complex has an isolated point. Indeed, we show that there is exactly one isolated point essentially only in the case where the monomial ideal is generated purely by powers of the variables. This allows us to recover Beintema's result that an Artinian monomial ring is Gorenstein if and only if it is a complete intersection. A key ingredient of the translation between the pure power result and Beintema's result is given by the one-to-one correspondence we show between the so-called \emph{truly isolated} points of our complex and the generators of the socle of the defining ideal. In another relation between the geometry of the complex and the algebra of the ring, we essentially give a correspondence between the nontrivial connected components of the complex and the factors of a fibre product representation of the ring. Finally, we explore algorithms for building survival complexes from specified isolated points. That is, we work to build the ring out of a description of the socle.