Hypersurface sections and \(\text{S}_2\)-modules
Let \(R\) be a local ring and \(M\) an equidimensional finitely generated module. Let \(I\) be the intersection of the embedded primes of \(M\) and let \(M^{un}:=M/H^0_I(M)\). Then \(\text{Ass}(M^{un})\) consists on the minimal primes of \(M\), hence \(M^{un}\) is unmixed (i.e., equidimensional and without embedded primes). The author's main result is the following one: Let \(S\) be a local Cohen-Macaulay ring, \(N\) an unmixed \(S\)-module, \(\underline x=x_1, \dots,x_n\) a sequence in \(S\) which is \(S/\text{Ann}(N)\)-regular and set \(R=S/ (\underline x)\), \(M=N/(\underline x)N\). Assume that \(N/(x_1, \dots,x_i)N\) is unmixed in codimension 1 for \(1\leq i\leq n\). If \(M^{un}\) satisfies Serre's condition \(\text{S}_2\) then \(M\) is unmixed (i.e., \(M=M^{un})\) and \(\underline x\) is \(N\)-regular. The proof follows an idea of \textit{C. Huneke} and \textit{B. Ulrich} [J. Algebr. Geom. 2, 487-505 (1993; Zbl 0808.14041)]. -- The applications of the result concern ideals, i.e., the case \(N=S/J\). Assume, in this case, that \(\underline x\) is also \(S\)-regular and put \(I=(J+(\underline x))/ (\underline x)\), i.e., \(M=R/I\). The author shows for example that: (a) If \(I\) is an unmixed ideal of positive height and generically a complete intersection, if \(S^n(J)\simeq J^n\) and if \(I^{(n)}=I^n\) then \(S^n(I)\simeq I^n\). (b) If \(I\) is a complete intersection in codimension 1, if \(J^{(n)}=J^n\) and if \(R/I^{(n)}\) satisfies \(\text{S}_2\) then \(I^{(n)}=I^n\).
scientific article; zbMATH DE number 1437095
scientific article