On the commutation of the test ideal with localization and completion (Q2716133)

Let \(R\) be a Noetherian ring containing a field of prime characteristic \(p>0\). Let \(N\hookrightarrow M\) be an inclusion of a submodule \(N\) in an \(R\)-module \(M\). The tight closure, \(N_M^*\), of \(N\) in \(M\) is the submodule of \(M\) consisting of all elements \(x\in M\) for which there exists \(c\in R\) not contained in any minimal prime of \(R\) such that \(c\otimes_Rx\in \text{Im}(R^{(e)}\otimes_RN\to R^{(e)}\otimes_RM)\) for all \(e\gg 0\). The test ideal of \(R\), denoted \(\tau\), consists of all elements \(c\in R\) that annihilate \(N^*_M/N\) for all finitely generated \(M\) and all submodules \(N\) of \(M\). NEWLINENEWLINENEWLINEThe second author of the present paper proved that the property of being a test ideal is preserved by localizations in case of a complete local Cohen-Macaulay ring [\textit{K. E. Smith}, Trans. Am. Math. Soc. 347, No. 9, 3453-3472 (1995; Zbl 0849.13003)]. We know that in a Gorenstein ring, the injective hull of the residue field is isomorphic to a local cohomology module, and therefore comes equipped with a natural action of Frobenius. This puts a natural \(R\{F\}\)-module structure on \(E\), where \(R\{F\}\) is the subring of \(\text{End}_{\mathbb Z}(R)\) generated by \(R\) and by the Frobenius operator \(F\). For Gorenstein local rings whose completion is a domain, it was shown that the \(R\{F\}\)-module \(E\) has a unique maximal proper submodule and its annihilator in \(R\) is the test ideal. Using this \(R\{F\}\)-module structure, the test ideal of a complete local Gorenstein domain was shown to behave well under localization. NEWLINENEWLINENEWLINEThe methods of \textit{K. E. Smith}'s paper [loc. cit.] break down for non-Gorenstein rings because, in general, there does not seem to be a natural \(R\{F\}\)-structure on \(E\). In this paper the authors consider all \(R\{F\}\)-module structures on \(E\). Let \(R\) be a reduced ring that is essentially of finite type over an excellent regular local ring of prime characteristic. Then it is shown that the test ideal of \(R\) commutes with localization.