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Koszul Gorenstein algebras from Cohen–Macaulay simplicial complexes
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D’Alì, Alessio and Venturello, Lorenzo (2023) Koszul Gorenstein algebras from Cohen–Macaulay simplicial complexes. International Mathematics Research Notices, 2023 (6). pp. 4998-5045. rnac003. doi:10.1093/imrn/rnac003 ISSN 1073-7928.
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Official URL: https://doi.org/10.1093/imrn/rnac003
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.
| Item Type: | Journal Article | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
| SWORD Depositor: | Library Publications Router | ||||||||
| Journal or Publication Title: | International Mathematics Research Notices | ||||||||
| Publisher: | Oxford University Press | ||||||||
| ISSN: | 1073-7928 | ||||||||
| Official Date: | March 2023 | ||||||||
| Dates: |
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| Volume: | 2023 | ||||||||
| Number: | 6 | ||||||||
| Page Range: | pp. 4998-5045 | ||||||||
| Article Number: | rnac003 | ||||||||
| DOI: | 10.1093/imrn/rnac003 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Access rights to Published version: | Restricted or Subscription Access |
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