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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
32/2018

Stanley-Reisner rings for symmetric simplicial complexes, $G$-semimatroids and Abelian arrangements

Alessio D'Ali and Emanuele Delucchi

Abstract

We extend the notion of face rings of simplicial complexes and simplicial posets to the case of finite-length simplicial posets with a group action. The action on the complex induces an action on the face ring, and we prove that the ring of invariants is isomorphic to the face ring of the quotient simplicial poset when the group action is translative (in the sense of Delucchi-Riedel). When the acted-upon poset is the independence complex of a semimatroid, the $h$-polynomial of the ring of invariants can be read off the Tutte polynomial of the associated $G$-semimatroid. We thus recover the classical theory in the case of trivial group actions on finite simplicial posets and, in the special case of central toric arrangements, our rings are isomorphic to those defined by Martino and by Lenz. We also describe a further condition on the group action ensuring that the topological Cohen-Macaulay property is preserved under quotients. In particular, we prove that the independence complex and the Stanley-Reisner ring of any Abelian arrangement are Cohen-Macaulay over every field. As a byproduct, we prove that posets of connected components (also known as posets of layers) of Abelian arrangements are (homotopically) Cohen-Macaulay.

Received:
Apr 23, 2018
Published:
Apr 24, 2018
MSC Codes:
13F55, 55U10, 06A11

Related publications

inJournal
2021 Repository Open Access
Alessio D'Alì and Emanuele Delucchi

Stanley-Reisner rings for symmetric simplicial complexes, G-semimatroids and Abelian arrangements

In: Journal of combinatorial algebra, 5 (2021) 3, pp. 185-236