Vanishing of differentials along ideals and non-archimedean approximation
Let \(R\) be a regular local ring, containing a perfect field \(k\), over which \(R\) is essentially of finite type, and let \(C(R/k)=\{x\in R:\delta (x)=0\) for all \(\delta\in \text{Der}_k (R)\}\). The authors prove that if \(R\) is a semi-local regular excellent and irreducible \(k\)-algebra such that \(R/R^p\) is finite (where \(p=\text{char}(k)>0)\), and \(I\subseteq R\) is an ideal, then there exists an integer \(l\) with the property that if \(x\in R\) with \(\delta(x)\in I^{n+l}\) for all \(\delta\in \text{Der}_k(R)\), then there exists a \(c\in C(R/k)\) with \(x-c\in I^n\). This provides a positive answer to a question raised by C. Huneke [see \textit{R. Fedder}, \textit{C. Huneke} and \textit{R. Hübl}, Proc. Am. Math. Soc. 108, No. 2, 319-325 (1990; Zbl 0691.13028)]. The solution for characteristic 0 was given by \textit{R. Hübl} [Proc. Am. Math. Soc. 127, No. 12, 3503-3511 (1999; Zbl 0938.13008)]. In the characteristic 0 case \(l\) can be bounded by a constant depending only on \(R\). But the situation in the present work is more difficult and the authors are able to find a uniform bound only with restrictions. The goal is to enable the use of these techniques to clarify or provide alternate proofs of known results and extend work to more general problems. See for example the algebraic formulation of the Kodaria vanishing theorem by \textit{C. Huneke} and \textit{K. Smith} [J. Reine Angew. Math. 484, 127-152 (1997; Zbl 0913.13003)].
scientific article; zbMATH DE number 1488549