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On the normality of a class of monomial ideals via the Newton polyhedron. (English) Zbl 1415.13008
Summary: Let \(I=\left\langle x_{1}^{a_{1}},\ldots,x_{n}^{a_{n}}\right\rangle \subset R=K[x_{1},\ldots,x_{n}]\) with \(a_{1},\ldots,a_{n}\) positive integers and \(K\) a field, and let \(J\) be the integral closure of \(I\). A criterion for the normality of \(J\) is developed. This criterion is used to show that \(J\) is normal if and only if the integral closure of the ideal \(\langle x_{1}^{b_{1}},\ldots,x_{n}^{b_{n}},\ldots,x_{r}^{b_{r}}\rangle \subset R[x_{n+1},\ldots,x_{r}]\) is normal, where \(b_{i}\in \left\{ a_{1},\ldots,a_{n}\right\} \) for all \(i\), this generalizes the work of the first author [Rocky Mt. J. Math. 39, No. 1, 1–9 (2009; Zbl 1166.13023)]. If \(l=lcm (a_{1},\ldots,a_{n})\) and the integral closure of \(\left\langle x_{1}^{a_{1}},\ldots,x_{n}^{a_{n}},x_{n+1}^{l}\right\rangle \subset R[x_{n+1}]\) is not normal, then we show that the integral closure of \(\left\langle x_{1}^{a_{1}},\ldots,x_{n}^{a_{n}},x_{n+1}^{s}\right\rangle \) is not normal for any \(s>l\). Also, we give a shorter proof of a main result of H. Coughlin [Classes of normal monomial ideals. Eugene, OR: University of Oregon (PhD Thesis) (2004)].

MSC:
13B22 Integral closure of commutative rings and ideals
06B10 Lattice ideals, congruence relations
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
Software:
Normaliz; SINGULAR
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