We investigate quotients by radical monomial ideals for which T2, the second cotangent cohomology module, vanishes. The dimension of the graded components of T2, 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 T2=0. We characterize the graded components of T2 when the simplicial complex is a uniform matroid. Finally, we show that T2 vanishes for all matroids of corank at most two and conjecture that all connected matroids with vanishing T2 are of corank at most two.
