The upper bound of the Dilworth number and the Rees number of Noetherian local rings with a Hilbert function
The Dilworth number and the Rees number of a local ring \(A\) are defined as follows: \[ d(A) = \sup\{\mu(I)|\;I\;\text{is an ideal of} A\}, \] \[ r(A) = \inf\{\ell(A/xA)|\;x\;\text{is a non-unit element of} A\}, \] where \(\mu(I)\) denotes the minimal number of generators of \(I\). These invariants were introduced and studied by \textit{J. Watanabe} in a series of papers [see e.g. Adv. Math. 76, No. 2, 194-199 (1989; Zbl 0703.13019)]. Note that \(d(A) < \infty\) if and only if \(\dim A \leq 1\). Given the Hilbert function of \(A\), this paper determines the possible maximum values of \(d(A)\) and \(r(A)\). Let \({\mathbf h} = h_0,h_1,h_2,\dots\) be a sequence of non-negative integers. Let \(k \) be an arbitrary field. Consider a polynomial ring \(R = k[X_1,\ldots,X_n]\), \(n = h_1\). Let \(M({\mathbf h})\) denote the union of the sets of the last \(h_i\) monomials of degree \(i\) in \(R\), \(i \geq 0\), with respect to the reverse lexicographic order. Let \(A(k,{\mathbf h})\) denote the quotient ring of \(R\) by the ideal generated by all monomials not contained in \(M({\mathbf h})\). Let \({\mathcal R}({\mathbf h})\) resp. \({\mathcal R}_0({\mathbf h})\) denote the set of local rings resp. local rings having residue fields of characteristic zero with Hilbert function \({\mathbf h}\). The sequence \({\mathbf h}\) is called an \(O\)-sequence if \({\mathbf h}\) is the Hilbert function of a local ring \(A\), i.e. \({\mathcal R}({\mathbf h}) \neq \emptyset\). In this case, \(A(k,{\mathbf h}) \in {\mathcal R}({\mathbf h})\). One can assign with \({\mathbf h}\) a number \(v({\mathbf h})\) which is finite if \(\dim A \leq 1\). For an \(O\)-sequence \({\mathbf h}\) the author proves the following: (i) \(d(A) \leq v({\mathbf h})\) for all \(A \in {\mathcal R}({\mathbf h})\); (ii) \(r(A) \leq v({\mathbf h})\) for all \(A \in {\mathcal R}_0({\mathbf h})\); (iii) \(d(A(k,{\mathbf h})) = r(A(k,{\mathbf h})) = v({\mathbf h})\). There is an example which shows that (ii) fails if we replace \({\mathcal R}_0({\mathbf h})\) by \({\mathcal R}({\mathbf h})\).
scientific article; zbMATH DE number 997643