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 Gröbner basis if and only if Delta is shellable. We give two applications: first, we construct quadratic Gorenstein F-algebras which 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.
Koszul Gorenstein algebras from Cohen-Macaulay simplicial complexes
D'Alì, Alessio;
In corso di stampa
Abstract
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 Gröbner basis if and only if Delta is shellable. We give two applications: first, we construct quadratic Gorenstein F-algebras which 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.
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