## Topological applications of Stanley-Reisner rings of simplicial complexes

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A. A. Aizenberg

Translated by: A. Martsinkovsky - Trans. Moscow Math. Soc.
**2012**, 37-65 - DOI: https://doi.org/10.1090/S0077-1554-2013-00200-9
- Published electronically: January 24, 2013
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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.## References

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## Bibliographic Information

**A. A. Aizenberg**- Affiliation: M. V. Lomonosov Moscow State University
- Email: ayzenberga@gmail.com
- 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
- © Copyright 2013 American Mathematical Society
- Journal: Trans. Moscow Math. Soc.
**2012**, 37-65 - MSC (2010): Primary 13F55; Secondary 55U10, 13H10
- DOI: https://doi.org/10.1090/S0077-1554-2013-00200-9
- MathSciNet review: 3184967