Big tight closure test elements for some non-reduced excellent rings (Q420690): Difference between revisions

In the theory of tight closure for commutative Noetherian rings of characteristic \(p > 0\), test elements play a central role. Recall that \(z\) is in the tight closure of an ideal \(I\) if there exists \(c \in R\), not in any minimal prime, such that \(c z^{p^e} \in I^{[p^e]}\) for all \(e \gg 0\) (here \(I^{[p^e]}\) is the ideal generated by the \(p^e\)th powers of elements of \(I\)). A test element is a choice of \(c\) that works for all \(z\) in the tight closure of \(I\).NEWLINENEWLINEIn the paper under review the author proves the following theorem, generalizing results of \textit{M. Hochster} and \textit{C. Huneke} [Trans. Am. Math. Soc. 346, No. 1, 1--62 (1994; Zbl 0844.13002)] to the non-reduced setting.NEWLINENEWLINETheorem. Suppose that \((R, \mathfrak{m})\) is an excellent local ring that is regular in codimension 0 (i.e., generically reduced). Then \(R\) has a test element. More generally if \(c\) is an element not in any minimal prime such that \(R_c\) is Gorenstein and weakly \(F\)-regular, then some power of \(c\) is a big test element.NEWLINENEWLINEIn fact, the author proves an even stronger statement. He shows that the test elements that are constructed are \textit{big test elements} (test elements that work for modules, including not necessarily finitely generated modules).