Koszul Gorenstein Algebras From Cohen-Macaulay Simplicial Complexes

We associate with every pure flag simplicial complex Delta a standard graded Gorenstein F-algebra R-Delta whose homological features are largely dictated by the combinatorics and topology of Delta. As our main result, we prove that the residue field F has a k-step linear R-Delta-resolution if and only if (Delta) satisfies Serre's condition (S-k) over F and that R-Delta is Koszul if and only if Delta is Cohen-Macaulay over F. Moreover, we show that R-Delta has a quadratic Grobner basis if and only if Delta is shellable. We give two applications: first, we construct quadratic Gorenstein F-algebras that are Koszul if and only if the characteristic of F is not in any prescribed set of primes. Finally, we prove that whenever R-Delta is Koszul the coefficients of its gamma-vector alternate in sign, settling in the negative an algebraic generalization of a conjecture by Charney and Davis.