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On Stanley-Reisner rings. (English) Zbl 0741.13006

Topics in algebra. Part 2: Commutative rings and algebraic groups, Pap. 31st Semester Class. Algebraic Struct., Warsaw/Poland 1988, Banach Cent. Publ. 26, Part 2, 57-70 (1990).
[For the entire collection see Zbl 0716.00007.]
Let \(S=k[x_ 1,\ldots,x_ n]\) be the polynomial ring in \(n\) variables over a field \(k\). For a homogeneous ideal \(I\) of \(S\) the graded \(k\)- algebra \(R=S/I\) is said to have \(t\)-linear resolution if \(I\) is minimally generated in degree \(t\) and all the syzygy modules of \(I\) are generated by linear syzygies. Let \(\Delta\) denote an abstract finite simplicial complex over the vertex set \(\{x_ 1,\ldots,x_ n\}\). Then the Stanley- Reisner ring of \(\Delta\) is defined by \(k[\Delta]=k[x_ 1,\ldots,x_ n]/I_ \Delta\), where \(I_ \Delta\) denotes the ideal generated by all monomials \(x_{i_ 1}\cdots x_{i_ r}\) such that \(\{x_{i_ 1},\ldots,x_{i_ r}\}\notin\Delta\). In the main results the author characterizes Stanley-Reisner rings with a 2-linear resolution in terms of graph-theoretical constructions of the underlying complex. Furthermore there are several results on numerical invariants of \(\Delta\) related to the Hilbert series of \(k[\Delta]\), provided \(k[\Delta]\) is a non-Cohen- Macaulay ring possessing a 2-linear resolution.
As a byproduct of his research the author gives a purely algebraic proof of D. E. Smith’s result [see Pac. J. Math. 141, No. 1, 165-196 (1960; Zbl 0686.13008)]: \(\text{depth}(k[\Delta])=1+\max\{i\in\mathbb{Z}:\) the \(i\)-skeleton \(\Delta^ i\) of \(\Delta\) has a Cohen-Macaulay Stanley- Reisner ring}.
Reviewer: P.Schenzel (Halle)

MSC:

13D25 Complexes (MSC2000)
55U10 Simplicial sets and complexes in algebraic topology
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13C13 Other special types of modules and ideals in commutative rings
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13D02 Syzygies, resolutions, complexes and commutative rings
06B05 Structure theory of lattices
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