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Free resolutions of simplicial posets. (English) Zbl 0882.06004
A simplicial poset is a finite poset with minimal element in which every interval is isomorphic to a Boolean lattice. This notion generalizes the face poset of a simplicial complex. Indeed, any simplicial poset may be realized as the face poset of a regular cell complex in which every cell is isomorphic to a simplex. R. Stanley defined a ring $$A_P$$ associated with a simplicial poset $$P$$ which generalizes the face ring (or Stanley-Reisner ring) of a simplicial complex. In the paper under review, the author extends several important calculations from face rings of simplicial complexes to face rings of simplicial posets. The Betti polynomial of module $$M$$ is the generating function for the ranks of the terms in a minimal free resolution of $$M$$. The first main result of the paper calculates the Betti polynomial of $$A_P$$ as a module over $$k[V]$$, the polynomial ring on the set $$V$$ of vertices of $$P$$. The result is expressed in terms of ordinary (topological) Betti numbers of subcomplexes of the realization of $$P$$. The proof follows the general outline of the proof in the simplicial complex case [M. Hochster, “Cohen-Macauley rings, combinatorics, and simplicial complexes”, in: Ring Theory II, Proc. Second Oklahoma Conference 1975, 171-223 (1977; Zbl 0351.13009)], but uses a refinement of the usual $${\mathbb{Z}}^n$$-grading of the Koszul complex ($$n=|V|$$). A similar, but more complicated analysis leads to the second main result, a calculation of the cohomology $$H^*(A_P):=H^*({\mathcal K}({\mathbf x}^\infty,A_P))$$ of a certain chain complex $${\mathcal K}({\mathbf x}^\infty,A_P)$$ determined by combinatorially defined local cohomology can be used to compute depth of modules. Thus the author is able to prove that (i) the depth of $$A_P$$ is a topological invariant, that is, depends only on the realization of $$P$$ as a topological space, and (ii) the depth of $$A_P$$ is equal to $$1+m$$, where $$m$$ is maximal such that the $$m$$-skeleton of the realization of $$P$$ is a Cohen-Macauley complex. These also generalize known properties of simplicial complexes. The article ends with a detailed example.

##### MSC:
 06A11 Algebraic aspects of posets
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##### References:
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