On Stanley-Reisner Rings with Minimal Betti Numbers

We classify simplicial complexes with a given number of vertices whose Stanley-Reisner ring has the minimal sum of Betti numbers.

The Betti numbers of the Stanley-Reisner rings of such kind of simplicial complexes are given by the binomial coefficients.

We obtain a topological characterization of these simplicial complexes K by showing that any full subcomplex of K is homotopy equivalent either to a point or to a sphere.

Moreover, we prove that such kind of simplicial complexes coincide with simplicial complexes having a minimal Taylor resolution.

This allows us to characterize these simplicial complexes in a purely combinatorial way.