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On Stanley-Reisner rings of reduction number one. (English) Zbl 1012.13007
Let \(\Delta\) denote a finite simplicial complex of dimension \(d\) on the vertex set \(\{x_1, \ldots, x_n\}.\) Let \(R = K[\Delta]\), \(K\) an infinite field, be the Stanley-Reisner ring of \(\Delta.\) The reduction number \(r(R)\) is the smallest number \(\rho\) for which there exists \(d+1\) linear forms \(g_1,\ldots,g_{d+1}\) such that \(R_{\rho +1} = (g_1, \ldots, g_{d+1})R_\rho.\) Each simplicial complex defines a simple one-dimensional non-directed graph. In terms of the coloring of this graph the authors give a sufficient criterion for the existence of a linear system of parameters of reduction exponent one. The corresponding class of Stanley-Reisner rings include the Cohen-Macaulay rings of minimal degree. In fact, the authors’ results generalize those of R. Fröberg [see: Topics in Algebra, Part 2: Commutative rings and algebraic groups, Pap. 31st Semester Class. Algebraic Struct., Warsaw 1988, Banach Cent. Publ. 26, 57-70 (1990; Zbl 0741.13006)].
MSC:
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
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