A simple proof of some generalized principal ideal theorems (Q2716094)

Let \(R\) be a Noetherian commutative ring and \(N\) a finitely generated \(R\)-module. Define the rank of \(N\) as being the maximum of the dimensions of the \(R_P/PR_P\)-vector spaces \(N_P/PN_P\), \(P\) ranging over the minimal prime ideals of \(R\). For each \(x\in N\), let \(N^*(x)=\{ f(x);\;f\in N^*\}\) be the order ideal of \(x\), \(N^*\) being the dual of \(N\). For a prime ideal \(Q\) of \(R\), let \(\mu_Q(R)\) denote the minimal number of generators of the \(R_Q\)-module \(N_Q\). The ``generalized principal ideal theorem'' [\textit{W. Bruns}, Proc. Am. Math. Soc. 83, 19-26 (1981; Zbl 0478.13006) and \textit{D. Eisenbud} and \textit{E. G. Evans}, Nagoya Math. J. 62, 41-53 (1976; Zbl 0324.13021)] says that if \((R,m)\) is local and \(x\in mN\), then \(\text{ht}(N^*(x))\leq \text{rank}(N)\). (The usual principal ideal theorem, that the height of a proper ideal generated by \(n\) elements is at most \(n\), is the case when \(N\) is free of rank \(n\).) NEWLINENEWLINENEWLINEIn the paper under review, the authors use symmetric algebras to simplify and strengthen this theorem. They prove that if \((R,m)\) is local and \(x\in mN\), then NEWLINE\[NEWLINE\dim(R/N^*(x))\geq \dim(R/Q)-\mu_Q(N)NEWLINE\]NEWLINE for every minimal prime ideal \(Q\) of \(R\); thus for every such \(Q\), NEWLINE\[NEWLINE\text{ht}(N^*(x))\leq \dim(R)- \dim(R/Q)+\mu_Q(N).NEWLINE\]NEWLINE The authors also give a simple proof and an extension of a result by \textit{M. Kwiecinski} [J. Algebra 194, No. 2, 378-382 (1997; Zbl 0907.13008)] which estimates the height of certain Fitting ideals of modules having equidimensional symmetric algebra.