Koszul Gorenstein algebras from Cohen-Macaulay simplicial complexes

TL;DR

This paper constructs and analyzes Gorenstein algebras associated with simplicial complexes, linking algebraic properties like Koszulness and quadratic Gr"obner bases to combinatorial and topological features of the complexes.

Contribution

It establishes new characterizations of Koszulness, Gorenstein properties, and quadratic Gr"obner bases in terms of simplicial complex conditions, and provides applications to algebraic and topological conjectures.

Findings

R_{ riangle} is Koszul iff riangle is Cohen-Macaulay.

Quadratic Gr"obner basis for R_{ riangle} iff riangle is shellable.

Constructs quadratic Gorenstein algebras with characteristic-dependent Koszulness.

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 Gr\"{o}bner 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.