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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



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