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arXiv:math.AC/0210278 Test ideals and base change problems in tight closure theory

Summary: arXiv:math.AC/0210278
Test ideals and base change problems in tight closure theory
Ian M. Aberbach and Florian Enescu
Abstract. Test ideals are an important concept in tight closure theory and their behavior
via at base change can be very diĘcult to understand. Our paper presents results regarding
this behavior under at maps with reasonably nice (but far from smooth) bers. This involves
analyzing, in depth, a special type of ideal of test elements, called the CS test ideal. Besides
providing new results, the paper also contains extensions of a theorem by G. Lyubeznik
and K. E. Smith on the completely stable test ideal and of theorems by F. Enescu and,
independently, M. Hashimoto on the behavior of F -rationality under at base change.
1. Introduction and terminology
Let R be a Noetherian commutative ring of positive prime characteristic p. Over the last
decade tight closure theory has played a tremendous role in understanding the structure
of R, [9{12], [16]. One compelling problem is how tight closure behaves under at base
change. This problem has multiple facets and it is intimately connected to the problem
of localization of tight closure, as shown by M. Hochster and C. Huneke in [11]. In fact,


Source: Aberbach, Ian - Department of Mathematics, University of Missouri-Columbia


Collections: Mathematics