Moment-angle complexes from simplicial posets

Zhi Lü; Taras Panov

Open Mathematics (2011)

  • Volume: 9, Issue: 4, page 715-730
  • ISSN: 2391-5455

Abstract

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We extend the construction of moment-angle complexes to simplicial posets by associating a certain T m-space Z S to an arbitrary simplicial poset S on m vertices. Face rings ℤ[S] of simplicial posets generalise those of simplicial complexes, and give rise to new classes of Gorenstein and Cohen-Macaulay rings. Our primary motivation is to study the face rings ℤ[S] by topological methods. The space Z S has many important topological properties of the original moment-angle complex Z K associated to a simplicial complex K. In particular, we prove that the integral cohomology algebra of Z S is isomorphic to the Tor-algebra of the face ring ℤ[S]. This leads directly to a generalisation of Hochster’s theorem, expressing the algebraic Betti numbers of the ring ℤ[S] in terms of the homology of full subposets in S. Finally, we estimate the total amount of homology of Z S from below by proving the toral rank conjecture for the moment-angle complexes Z S.

How to cite

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Zhi Lü, and Taras Panov. "Moment-angle complexes from simplicial posets." Open Mathematics 9.4 (2011): 715-730. <http://eudml.org/doc/269523>.

@article{ZhiLü2011,
abstract = {We extend the construction of moment-angle complexes to simplicial posets by associating a certain T m-space Z S to an arbitrary simplicial poset S on m vertices. Face rings ℤ[S] of simplicial posets generalise those of simplicial complexes, and give rise to new classes of Gorenstein and Cohen-Macaulay rings. Our primary motivation is to study the face rings ℤ[S] by topological methods. The space Z S has many important topological properties of the original moment-angle complex Z K associated to a simplicial complex K. In particular, we prove that the integral cohomology algebra of Z S is isomorphic to the Tor-algebra of the face ring ℤ[S]. This leads directly to a generalisation of Hochster’s theorem, expressing the algebraic Betti numbers of the ring ℤ[S] in terms of the homology of full subposets in S. Finally, we estimate the total amount of homology of Z S from below by proving the toral rank conjecture for the moment-angle complexes Z S.},
author = {Zhi Lü, Taras Panov},
journal = {Open Mathematics},
keywords = {Moment-angle complex; Simplicial poset; Simplicial complex; Torus action; Face ring; Stanley-Reisner ring; Toral rank},
language = {eng},
number = {4},
pages = {715-730},
title = {Moment-angle complexes from simplicial posets},
url = {http://eudml.org/doc/269523},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Zhi Lü
AU - Taras Panov
TI - Moment-angle complexes from simplicial posets
JO - Open Mathematics
PY - 2011
VL - 9
IS - 4
SP - 715
EP - 730
AB - We extend the construction of moment-angle complexes to simplicial posets by associating a certain T m-space Z S to an arbitrary simplicial poset S on m vertices. Face rings ℤ[S] of simplicial posets generalise those of simplicial complexes, and give rise to new classes of Gorenstein and Cohen-Macaulay rings. Our primary motivation is to study the face rings ℤ[S] by topological methods. The space Z S has many important topological properties of the original moment-angle complex Z K associated to a simplicial complex K. In particular, we prove that the integral cohomology algebra of Z S is isomorphic to the Tor-algebra of the face ring ℤ[S]. This leads directly to a generalisation of Hochster’s theorem, expressing the algebraic Betti numbers of the ring ℤ[S] in terms of the homology of full subposets in S. Finally, we estimate the total amount of homology of Z S from below by proving the toral rank conjecture for the moment-angle complexes Z S.
LA - eng
KW - Moment-angle complex; Simplicial poset; Simplicial complex; Torus action; Face ring; Stanley-Reisner ring; Toral rank
UR - http://eudml.org/doc/269523
ER -

References

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