Tight closure in non-excellent rings
Introduced by Hochster and Huneke, the tight closure theory is an important tool for studying excellent rings of equal characteristic. The paper under review is a study of the tight closure in non-excellent local rings of prime characteristic \(p\). The author provides examples showing that important properties such as ``colon capturing'' and ``persistence'', which make tight closure such a powerful tool in excellent rings, do not necessarily hold in the non-excellent case. If \(I^{\ast}\) denotes the tight closure of the ideal \(I\), the colon capturing property means that for \(x_1,\dots, x_n\) parameters in \(R\), we have \(((x_1,\dots, x_{n-1}): x_n) \subseteq (x_1,\dots, x_{n-1})^{\ast}\), while persistence means that \(\phi(I^{\ast}) \subseteq (\phi(I)S)^{\ast}\) whenever \(\phi:R \to S\) is a ring homomorphism. First it is noted that in the ring from Example 2 in the Appendix of \textit{M. Nagata} [Local rings. New York and London: Interscience Publishers (1962; Zbl 0123.03402)], which is not universally catenarian, the colon capturing of tight closure fails. Moreover, the author provides examples of universally catenarian rings \(R\) where the same property fails. As a consequence, this shows that the plus closure \(R^{+}\) need not be Cohen-Macaulay. Finally, the author shows the failure of the persistence of tight closure in non-excellent rings by giving an example of a four dimensional unique factorization domain and a prime element \(x \in R\) such that \(\pi(I^{\ast}) \not\subseteq (\pi(I))^{\ast}\), where \(\pi : R \to R/xR\) is the natural projection. To overcome the difficulties caused by the failure of persistence and colon capturing in the non-excellent case, the author introduces a new definition of tight closure for local rings \(R\) of prime characteristic \(p\). More precisely, if \(N\) is a submodule of an \(R\)-module \(M\) and \(x \in M\), then \(x\) is said to be in the general tight closure of \(N\) if \(x \otimes 1 \in (N \otimes \widehat{R})^{\ast}\) where \(\widehat{R}\) is the completion of \(R\) and \((N \otimes \widehat{R})^{\ast}\) is the usual tight closure of \(N \otimes \widehat{R} \) in \(M \otimes \widehat{R}\). Whenever \(R\) is an excellent ring, it is noted that the general tight closure coincides with the usual tight closure. Moreover, it is shown that this new closure has many of the properties that the classical tight closure has in the excellent case, including colon capturing and persistence. As a consequence, by using the general tight closure in the general case, the author extends some previously known results about tight closure in excellent rings.
scientific article; zbMATH DE number 5768892
scientific article