Simplicial complexes and matroids with vanishing $T^2$

TL;DR

This paper explores the conditions under which the second cotangent cohomology module $T^2$ vanishes for quotients by radical monomial ideals, linking algebraic properties to combinatorial structures like simplicial complexes and matroids.

Contribution

It provides a complete characterization of when $T^2$ vanishes for certain complexes and matroids, including a full list for one-dimensional complexes and results for matroids of low corank.

Findings

Vanishing of $T^2$ depends solely on the combinatorics of the simplicial complex.

Complete characterization of $T^2=0$ for one-dimensional complexes.

Proved $T^2$ vanishes for all matroids of corank at most two.

Abstract

We investigate quotients by radical monomial ideals for which $T^2$, the second cotangent cohomology module, vanishes. The dimension of the graded components of $T^2$, and thus their vanishing, depends only on the combinatorics of the corresponding simplicial complex. We give both a complete characterization and a full list of one dimensional complexes with $T^2=0$. We characterize the graded components of $T^2$ when the simplicial complex is a uniform matroid. Finally, we show that $T^2$ vanishes for all matroids of corank at most two and conjecture that all connected matroids with vanishing $T^2$ are of corank at most two.