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.

D'Ali, A., Venturello, L. (2023). Koszul Gorenstein Algebras From Cohen-Macaulay Simplicial Complexes. INTERNATIONAL MATHEMATICS RESEARCH NOTICES, 2023(6), 4998-5045 [10.1093/imrn/rnac003].

Koszul Gorenstein Algebras From Cohen-Macaulay Simplicial Complexes

Venturello L.
2023-01-01

Abstract

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.
2023
D'Ali, A., Venturello, L. (2023). Koszul Gorenstein Algebras From Cohen-Macaulay Simplicial Complexes. INTERNATIONAL MATHEMATICS RESEARCH NOTICES, 2023(6), 4998-5045 [10.1093/imrn/rnac003].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11365/1256095