Algebraically rigid simplicial complexes and graphs

TL;DR

This paper investigates algebraically rigid and inseparable simplicial complexes and graphs, characterizing their properties through Stanley-Reisner rings and applying the theory to letterplace and edge ideals.

Contribution

It introduces the concepts of algebraic rigidity and inseparability for simplicial complexes, providing classifications and applying the theory to specific ideals.

Findings

Identified classes of algebraically rigid simplicial complexes

Established relationships between rigidity and inseparability

Applied theory to letterplace and edge ideals of graphs

Abstract

We call a simplicial complex algebraically rigid if its Stanley-Reisner ring admits no nontrivial infinitesimal deformations, and call it inseparable if does not allow any deformation to other simplicial complexes. Algebraically rigid simplicial complexes are inseparable. In this paper we study inseparability and rigidity of Stanley-Reisner rings, and apply the general theory to letterplace ideals as well as to edge ideals of graphs. Classes of algebraically rigid simplicial complexes and graphs are identified.