# Stanley-Reisner rings of simplicial complexes with a free action by an abelian group

## TL;DR

This paper explores the algebraic and topological properties of Stanley-Reisner rings associated with simplicial complexes that have a free abelian group action, extending classical results and providing new bounds for special cases.

## Contribution

It refines Hochster's local cohomology description for complexes with free abelian group actions and generalizes bounds on the h-vector for Buchsbaum and Cohen-Macaulay complexes.

## Key findings

- Refined Hochster's local cohomology modules description.
- Extended Schenzel's Hilbert series calculations to Buchsbaum complexes.
- Provided lower bounds on h-vectors for complexes with free cyclic group actions.

## Abstract

We consider simplicial complexes admitting a free action by an abelian group. Specifically, we establish a refinement of the classic result of Hochster describing the local cohomology modules of the associated Stanley--Reisner ring, demonstrating that the topological structure of the free action extends to the algebraic setting. If the complex in question is also Buchsbaum, this new description allows for a specialization of Schenzel's calculation of the Hilbert series of some of the ring's Artinian reductions. In further application, we generalize to the Buchsbaum case the results of Stanley and Adin that provide a lower bound on the $h$-vector of a Cohen-Macaulay complex admitting a free action by a cyclic group of prime order.