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

The first Cotangent Cohomology Module for Matroids