Alexandru Constantinescu
Freie Universität Berlin, Germany & Simion Stoilow Institute of Mathematics of the Romanian Academy, Romania

The first and the second cotangent cohomology modules ($T^1$ and $T^2$) of a commutative ring $R$ control the algebraic deformations of the associated affine scheme. The first module is in bijection with the first order deformations (i.e.deformations with parameter space $\mathbb{C}[\varepsilon]/(\varepsilon)^2$) and the second module contains the obstructions to lifting such deformations to larger parameter spaces. Any grading on $R$ is inherited by $T^1$ and $T^2$. In such a setting, the degree zero components control the algebraic deformations of the associated projective scheme. The main result of my talk states that $T^1$ a Stanley-Reisner completely characterizes if the underlying simplicial complex is a matroid. Furthermore, in such cases, one can completely recover the matroid from the multigraded components of $T^1$. This is far from being true for simplicial complexes in general. I will also present a simple formula for computing $T^1$ for a matroid, and some partial results on $T^2$ for matroids.