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 which 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.

v2: 36 pages, 2 figures, 2 tables. Added Section 8 (about an Artinian reduction of R_Delta), improved the introduction, made some minor changes throughout

References in corpus (6)

• Gamma-positivity in combinatorics and geometry
• Spheres arising from multicomplexes
• The Koszul property for spaces of quadrics of codimension three
• Resolutions of co-letterplace ideals and generalizations of Bier spheres
• Regularity, singularities and $h$-vector of graded algebras
• Minimal flag triangulations of lower-dimensional manifolds