On a generalization of test ideals

By N. Hara, S. Takagi. Nagoya mathematical journal, 2002

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Abstract The test ideal τ(R) of a ring R of prime characteristic is an important object in the theory of tight closure. In this paper, we study a generalization of the test ideal, which is the ideal τ(at) associated to a given ideal a with rational exponent t ≥ 0. We first prove a key lemma of this paper (Lemma 2.1), which gives a characterization of the ideal τ(at). As applications of this key lemma, we generalize the preceding results on the behavior of the test ideal τ(R). Moreover, we prove an analogue of so-called Skoda’s theorem, which is formulated algebraically via adjoint ideals by Lipman in his proof of the “modified Briançon-Skoda theorem.”

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0c9504220d4f10682aa0be007d18203c6f862ca0

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Citations in the corpus

Integral closure of ideals, rings, and modules (2006, methodology): “It has been further studied in Hara [105], [106]; Smith [275]; Takagi [287]; Hara and Takagi [107]; and Hara and Watanabe [109].”
Discreteness and rationality of F-jumping numbers on singular varieties (2009, unknown): “See [16,35] for additional details.”
Formulas for multiplier ideals on singular varieties (2004, methodology): “A criterion for the existence of a t -test elements given in [ HT , Corollary 2.4] also works for “test elements for 0 ∗a”