Double cohomology of moment-angle complexes

Limonchenko, Ivan; Panov, Taras; Song, Jongbaek; Stanley, Donald

doi:10.1016/j.aim.2023.109274

Cited by 2 publications

(2 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 3.11. A recently introduced invariant, termed the double cohomology of a moment-angle complex, is defined in [28] as the homology of a chain complex of the form H * (Z K ), d or, equivalently, as the second page of one of the spectral sequences associated to a double complex k[K] ⊗ Λ(u 1 , . .…”

Section: Higher Cohomology Operationsmentioning

confidence: 99%

Untitled

2024

Trans. Amer. Math. Soc. Ser. B

View full text Add to dashboard Cite

A fundamental result in toric topology identifies the cohomology ring of the moment-angle complex 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.

show abstract

Section: Higher Cohomology Operationsmentioning

confidence: 99%

Untitled

2024

Trans. Amer. Math. Soc. Ser. B

View full text Add to dashboard Cite

show abstract

Section: Higher Cohomology Operationsmentioning

confidence: 99%

Cohomology operations for moment-angle complexes and resolutions of Stanley–Reisner rings

Amelotte,

Briggs

2024

Trans. Amer. Math. Soc. Ser. B

View full text Add to dashboard Cite

A fundamental result in toric topology identifies the cohomology ring of the moment-angle complex Z K \mathcal {Z}_K associated to a simplicial complex K K with the Koszul homology of the Stanley–Reisner ring of K 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.

show abstract

Double cohomology of moment-angle complexes

Cited by 2 publications

References 5 publications

Untitled

Untitled

Cohomology operations for moment-angle complexes and resolutions of Stanley–Reisner rings

Contact Info

Product

Resources

About