Comparing the face rings of a boolean complex and its barycentric subdivision

TL;DR

This paper investigates the relationship between face rings of boolean complexes and their barycentric subdivisions, establishing isomorphisms under certain algebraic conditions and generalizing tools for their construction.

Contribution

It demonstrates that face rings are generally not isomorphic but provides conditions and explicit methods for isomorphism in Cohen-Macaulay cases, extending Garsia's tools.

Findings

Face rings are not always isomorphic after barycentric subdivision.

Isomorphism holds for Cohen-Macaulay complexes in certain characteristics.

Provides explicit construction of isomorphisms using generalized Garsia tools.

Abstract

We consider the relationship between the Stanley-Reisner ring (a.k.a. face ring) of a simplicial or boolean complex $\Delta$ and that of its barycentric subdivision. These rings share a distinguished parameter subring. S. Murai asked if they are isomorphic, equivariantly with respect to the automorphism group $\operatorname{Aut}(\Delta)$, as modules over this parameter subring. We show that, in general, the answer is no, but for Cohen-Macaulay complexes in characteristic coprime to $|\operatorname{Aut}(\Delta)|$, it is yes, and we give an explicit construction of an isomorphism. To give this construction, we adapt and generalize a pair of tools introduced by A. Garsia in 1980. The first one transfers bases from a Stanley-Reisner ring to closely related rings of which it is a Gr\"obner degeneration, and the second identifies bases to transfer.