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Cohomology operations for moment-angle complexes and resolutions of Stanley–Reisner rings
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Title: | Cohomology operations for moment-angle complexes and resolutions of Stanley–Reisner rings |
Authors: | Amelotte, S Briggs, B |
Item Type: | Journal Article |
Abstract: | A fundamental result in toric topology identifies the cohomology ring of the moment-angle complex ZK associated to a simplicial complex K with the Koszul homology of the Stanley–Reisner ring of K. By studying cohomology operations induced by the standard torus action on the moment-angle complex, we extend this to a topological interpretation of the minimal free resolution of the Stanley–Reisner ring. The exterior algebra module structure in cohomology induced by the torus action recovers the linear part of the minimal free resolution, and we show that higher cohomology operations induced by the action (in the sense of Goresky–Kottwitz–MacPherson [Invent. Math. 131 (1998), pp. 25–83]) can be assembled into an explicit differential on the resolution. Describing these operations in terms of Hochster’s formula, we recover and extend a result due to Katth¨an [Mathematics 7 (2019), no. 7, p. 605]. We then apply all of this to study the equivariant formality of torus actions on moment-angle complexes. For these spaces, we obtain complete algebraic and combinatorial characterisations of which subtori of the naturally acting torus act equivariantly formally. |
Issue Date: | 2024 |
Date of Acceptance: | 1-Apr-2024 |
URI: | http://hdl.handle.net/10044/1/114030 |
DOI: | 10.1090/btran/181 |
ISSN: | 2330-0000 |
Publisher: | American Mathematical Society |
Start Page: | 826 |
End Page: | 862 |
Journal / Book Title: | Transactions of the American Mathematical Society. Series B |
Volume: | 11 |
Issue: | 25 |
Copyright Statement: | © 2024 by the author(s) under Creative Commons Attribution 3.0 License (CC BY 3.0) |
Publication Status: | Published |
Online Publication Date: | 2024-04-26 |
Appears in Collections: | Pure Mathematics Mathematics |
This item is licensed under a Creative Commons License