Cohomology operations for moment-angle complexes and resolutions of Stanley–Reisner rings
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- by Steven Amelotte and Benjamin Briggs;
- Trans. Amer. Math. Soc. Ser. B 11 (2024), 826-862
- DOI: https://doi.org/10.1090/btran/181
- Published electronically: April 26, 2024
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Abstract:
A fundamental result in toric topology identifies the cohomology ring of the moment-angle complex $\mathcal {Z}_K$ 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 Katthän [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.References
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Bibliographic Information
- Steven Amelotte
- Affiliation: Department of Mathematics, Western University, London, Ontario N6A 5B7, Canada
- MR Author ID: 1262469
- ORCID: 0000-0002-1357-8974
- Email: samelot@uwo.ca
- Benjamin Briggs
- Affiliation: Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5 DK-2100 Copenhagen Ø, Denmark
- MR Author ID: 1281297
- ORCID: 0000-0003-0003-9493
- Email: bpb@math.ku.dk
- Received by editor(s): May 24, 2023
- Received by editor(s) in revised form: October 31, 2023
- Published electronically: April 26, 2024
- Additional Notes: For part of this work, the first author was hosted by the Institute for Computational and Experimental Research in Mathematics in Providence, RI, supported by the National Science Foundation under Grant No. 1929284. The second author was funded by the European Union under the Grant Agreement no. 101064551 (Hochschild).
- © Copyright 2024 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)
- Journal: Trans. Amer. Math. Soc. Ser. B 11 (2024), 826-862
- MSC (2020): Primary 13F55, 57S12, 55U10
- DOI: https://doi.org/10.1090/btran/181
- MathSciNet review: 4739194