Koszul Gorenstein Algebras From Cohen-Macaulay Simplicial Complexes

Abstract We associate with every pure flag simplicial complex $\Delta $ a standard graded Gorenstein $\mathbb {F}$-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 $\mathbb {F}$ has a $k$-step linear $R_{\Delta }$-resolution if and only if $\Delta $ satisfies Serre’s condition $(S_k)$ over $\mathbb {F}$ and that $R_{\Delta }$ is Koszul if and only if $\Delta $ is Cohen–Macaulay over $\mathbb {F}$. Moreover, we show that $R_{\Delta }$ has a quadratic Gröbner basis if and only if $\Delta $ is shellable. We give two applications: first, we construct quadratic Gorenstein $\mathbb {F}$-algebras that are Koszul if and only if the characteristic of $\mathbb {F}$ 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.