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arXiv · 2507.20037

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

Ben Blum-Smith·Sophie Marques
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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 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.

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Ben Blum-Smith, Sophie Marques. 2025-07-26. Comparing the face rings of a boolean complex and its barycentric subdivision. https://arxiv.org/abs/2507.20037

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