  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.
