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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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##### References:
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