A variant of Wang's theorem
In this nicely written paper, the authors use characteristic \(p\) techniques to give a new variant, as the title suggests, of \textit{H. J. Wang}'s theorem [Math. Z. 256, 749--756 (2007; Zbl 1123.13018)]. Wang's theorem states that for a Cohen-Macaulay local ring \((R, \mathfrak{m})\) of dimension \(d \geq 2\) with \(d \geq 3\) if \(R\) is regular, if \(J \subseteq \mathfrak{m}^s\) for some \(s \geq 2\), then \(L:=J: \mathfrak{m}^s \subseteq \mathfrak{m}^s\) and \(L^2=JL\). The authors' interest in Wang's theorem stems from finding the Goto number of a parameter ideal in a more general setting. The Goto number of a parameter ideal \(J\) in a Noetherian local ring \((R,\mathfrak{m})\) was defined as \(g(J):=\text{max}\{t \in \mathbb{N} \mid J:\mathfrak{m}^t \text{ is integral over } J\}\) by \textit{W. Heinzer} and \textit{I. Swanson} [J. Algebra 321, 152--166 (2009; Zbl 1166.13018)]. The authors' first result is to show that in a quasi-unmixed Noetherian local ring or characteristic \(p\), for any subsystem of parameters \(J=(a_1,\ldots, a_m)\), the integral closure of \(J\) is equal to the \(J^{m-1}\)-tight closure of \(J\). They use this fact along with known results on \(\mathfrak{a}^t\)-test ideals to show that \(J:\overline{J}=J+\tau(J^{d-1})\) for any parameter ideal in a complete or \(F\)-finite normal Gorenstein local ring. Relying on work of \textit{N. Hara} and \textit{K. Yoshida} [Trans. Am. Math. Soc. 355, No. 8, 3143--3174 (2003; Zbl 1028.13003)] they then prove some Skoda-type theorems pertaining to the \(\mathfrak{m}^t\)-test ideals for \(t \geq d-1\) where \(d\) is the dimension of the ring, their strongest result being \(\tau(\mathfrak{m}^t) \subseteq \mathfrak{m}^{t-d+3}\) when \((R, \mathfrak{m})\) is a complete Gorenstein local ring with multiplicity \(e(R) \geq 3\). They then proceed to prove their new variant of Wang's theorem which states that if \(R\) is a Cohen-Macaulay ring of dimension \(d\geq 2\) and \(J \subseteq \mathfrak{m}^s\) is a parameter ideal for \(s \geq 2\), then \(J:\mathfrak{m}^{(d-1)(s-1)}\) is integral over \(J\) and if \(R\) is not regular then \(J:\mathfrak{m}^{(d-1)(s-1)+1}\) is integral over \(J\). Examples are provided both to illustrate that indeed in the regular case, \((d-1)(s-1)\) is indeed the Goto number of the parameter ideal \((x_1^s,\ldots, x_d^s)\) living in the power series ring \(kx_1, \ldots, x_d\) and in the non-regular case \((d-1)(s-1)+1\) need not be the Goto number of a parameter ideal in \(\mathfrak{m}^s\). Using the fact that after reducing to characteristic \(p\), the multiplier ideal \(\mathcal{J}(\mathfrak{a})=\tau(\mathfrak{a})\) they extend their results to the equicharacteristic 0 setting. In their final section, the authors present several nice questions about possible extensions of their variant of Wang's theorem providing several examples and some partial answers.
scientific article; zbMATH DE number 6155227
scientific article