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Topological applications of Stanley-Reisner rings of simplicial complexes

Author: A. A. Aizenberg
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva.
Journal: Trans. Moscow Math. Soc. 2012, 37-65
MSC (2010): Primary 13F55; Secondary 55U10, 13H10
Published electronically: January 24, 2013
MathSciNet review: 3184967
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Abstract: Methods of commutative and homological algebra yield information on the Stanley-Reisner ring $ \Bbbk [K]$ of a simplicial complex $ K$. Consider the following problem: describe topological properties of simplicial complexes with given properties of the ring $ \Bbbk [K]$. It is known that for a simplicial complex $ K=\partial P^*$, where $ P^*$ is a polytope dual to the simple polytope $ P$ of dimension $ n$, the depth of $ \operatorname {depth}\Bbbk [K]$ equals $ n$. A recent construction allows us to associate a simplicial complex $ K_P$ to any convex polytope $ P$. As a consequence, one wants to study the properties of the rings  $ \Bbbk [K_P]$. In this paper, we report on the obtained results for both of these problems. In particular, we characterize the depth of $ \Bbbk [K]$ in terms of the topology of links in the complex $ K$ and prove that  $ \operatorname {depth}\Bbbk [K_P] = n$ for all convex polytopes $ P$ of dimension $ n$. We obtain a number of relations between bigraded betti numbers of the complexes $ K_P$. We also establish connections between the above questions and the notion of a $ k$-Cohen-Macaulay complex, which leads to a new filtration on the set of simplicial complexes.

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Additional Information

A. A. Aizenberg
Affiliation: M. V. Lomonosov Moscow State University

Keywords: Stanley-Reisner ring, Reisner theorem, depth, Cohen-Macaulay ring, Gorenstein complex, moment-angle complex, nerve-complex
Published electronically: January 24, 2013
Additional Notes: This work was supported by the grants RFFI 11-01-00694-a and 12-01-92104-YaF_a
Article copyright: © Copyright 2013 American Mathematical Society