On the Stanley-Reisner ideal of an expanded simplicial complex

TL;DR

This paper investigates how algebraic and combinatorial properties of a simplicial complex are preserved under expansions, focusing on Cohen-Macaulayness, Buchsbaum properties, and homological invariants of Stanley-Reisner ideals.

Contribution

It establishes that key properties like Cohen-Macaulayness and $k$-decomposability are equivalent for a complex and all its expansions, and compares invariants like regularity and projective dimension.

Findings

Properties such as Cohen-Macaulayness are preserved under expansions.

Homological invariants like regularity are compared between a complex and its expansions.

The study links algebraic and combinatorial properties through expansions.

Abstract

Let $\Delta$ be a simplicial complex. We study the expansions of $\Delta$ mainly to see how the algebraic and combinatorial properties of $\Delta$ and its expansions are related to each other. It is shown that $\Delta$ is Cohen-Macaulay, sequentially Cohen-Macaulay, Buchsbaum or $k$-decomposable, if and only if an arbitrary expansion of $\Delta$ has the same property. Moreover, some homological invariants like the regularity and the projective dimension of the Stanley-Reisner ideals of $\Delta$ and those of their expansions are compared.