The first Cotangent Cohomology Module for Matroids

TL;DR

This paper provides a combinatorial formula for the first cotangent cohomology module of Stanley-Reisner rings of matroids, offering a new characterization and invariant for matroids based on cohomology.

Contribution

It introduces a combinatorial formula for T^1 of matroid Stanley-Reisner rings and characterizes matroids via cohomology bounds and invariants.

Findings

Derived a combinatorial formula for T^1 of matroids

Established bounds for T^1 components in simplicial complexes

Characterized matroids using cohomology as a complete invariant

Abstract

We find a combinatorial formula which computes the first cotangent cohomology module of Stanley-Reisner rings associated to matroids. For arbitrary simplicial complexes we provide upper bounds for the dimensions of the multigraded components of T^1. For specific degrees we prove that these bounds are reached if and only if the simplicial complex is a matroid, obtaining thus a new characterization for matroids. Furthermore, the graded first cotangent cohomology turns out to be a complete invariant for nondiscrete matroids.