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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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[1] Björner, A., Posets, regular CW complexes and Bruhat order, Europ. J. combin., 5, 7-16, (1984) · Zbl 0538.06001
[2] Björner, A.; Hibi, T., Betti numbers of Buchsbaum complexes, Math. scand., 67, 193-196, (1990) · Zbl 0727.55011
[3] A. Duval, 1991, Simplicial Posets:f, Massachusetts Institute of Technology
[4] Garsia, A.; Stanton, D., Group actions on stanley – reisner rings and invariants of permutation groups, Adv. in math., 51, 107-201, (1984) · Zbl 0561.06002
[5] Hibi, T., Quotient algebras of stanley – reisner rings and local cohomology, J. algebra, 140, 336-343, (1991) · Zbl 0761.55015
[6] M. Hochster, 1977, Cohen-Macaulay rings, combinatorics, and simplicial complexes, Ring Theory II: Proc. of the Second Oklahoma Conference, 171, 223, Dekker, New York
[7] Lang, S., Algebra, (1984), Addison-Wesley Menlo Park
[8] Munkres, J., Elements of algebraic topology, (1984), Benjamin/Cummings Menlo Park · Zbl 0673.55001
[9] Munkres, J., Topological results in combinatorics, Mich. math. J., 31, 113-128, (1984) · Zbl 0585.57014
[10] Reisner, G., Cohen – macaulay quotients of polynomial rings, Adv. in math., 21, 30-49, (1976) · Zbl 0345.13017
[11] Smith, D., On the cohen – macaulay property in commutative algebra and simplicial topology, Pac. J. math., 141, 165-196, (1990) · Zbl 0686.13008
[12] Spanier, E., Algebraic topology, (1966), McGraw-Hill New York · Zbl 0145.43303
[13] Stanley, R., Combinatorics and commutative algebra, Progress in mathematics, 41, (1995), Birkhäuser Boston
[14] Stanley, R., Enumerative combinatorics, (1986), Wadsworth & Brooks/Cole Monterey
[15] Stanley, R., fh, J. pure appl. algebra, 71, 319-331, (1991)
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