Koszul Gorenstein Algebras From Cohen-Macaulay Simplicial Complexes

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