Comparing the face rings of a boolean complex and its barycentric subdivision
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909708463374336 |
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| author | Blum-Smith, Ben Marques, Sophie |
| author_facet | Blum-Smith, Ben Marques, Sophie |
| contents | We consider the relationship between the Stanley-Reisner ring (a.k.a. face ring) of a simplicial or boolean complex $Δ$ 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 $\operatorname{Aut}(Δ)$, as modules over this parameter subring. We show that, in general, the answer is no, but for Cohen-Macaulay complexes in characteristic coprime to $|\operatorname{Aut}(Δ)|$, 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 Gröbner degeneration, and the second identifies bases to transfer. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_20037 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Comparing the face rings of a boolean complex and its barycentric subdivision Blum-Smith, Ben Marques, Sophie Commutative Algebra Combinatorics 13F55, 05E18, 05E40, 13C14, 13A50, 13C70, 05E45 We consider the relationship between the Stanley-Reisner ring (a.k.a. face ring) of a simplicial or boolean complex $Δ$ 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 $\operatorname{Aut}(Δ)$, as modules over this parameter subring. We show that, in general, the answer is no, but for Cohen-Macaulay complexes in characteristic coprime to $|\operatorname{Aut}(Δ)|$, 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 Gröbner degeneration, and the second identifies bases to transfer. |
| title | Comparing the face rings of a boolean complex and its barycentric subdivision |
| topic | Commutative Algebra Combinatorics 13F55, 05E18, 05E40, 13C14, 13A50, 13C70, 05E45 |
| url | https://arxiv.org/abs/2507.20037 |