Monomiale Ideale und mengentheoretische vollständige Durchschnitte. (Monomial ideals and set theoretical complete intersections) (Q1109835)

Let \(R=k[X_ 1,...,X_ n]\) be the polynomial ring in n indeterminates over the field k of arbitrary characteristic. The author considers ideals \(I\subset R\) which are generated by monomials and proves: If dim(R/I)\(\geq 1\) then the \(arithmeticrank\quad r\) of I, i.e. the smallest number r of polynomials \(f_ 1,...,f_ r\in R\) such that \(Rad(I)=Rad(f_ 1,...,f_ r)\), is at most n-1. Especially: If \(\dim (R/I)=1\), then the arithmetical rank of I is \(n=1\), i.e. I is a set-theoretical complete intersection. The last result for the case \(char(k)>0\) is contained in a paper by \textit{G. Lyubeznik} [``Some theorems on set theoretic intersections'' (Preprint); cf. also his thesis (Columbia Univ. 1984) and his paper in J. Algebra 87, 105-112 (1984; Zbl 0575.14041)] in which he shows this result for arbitrary ideals \(I\subset R\) thus generalizing the well-known result of \textit{R. C. Cowsik} and \textit{M. V. Nori} [Invent. Math. 45, 111-114 (1978; Zbl 0385.14015)]. In the case of characteristic zero Lyubeznik gives this result only for the case where I is locally a complete intersection. The author's proofs are straightforward calculations.