A generalization of tight closure and multiplier ideals

By N. Hara, KEN-ICHI Yoshida. 2002

226 citations recorded by the source corpus.

We introduce a new variant of tight closure associated to any fixed ideal a, which we call α-tight closure, and study various properties thereof. In our theory, the annihilator ideal r(a) of all α-tight closure relations, which is a generalization of the test ideal in the usual tight closure theory, plays a particularly important role. We prove the correspondence of the ideal τ(a) and the multiplier ideal associated to a (or, the adjoint of a in Lipman's sense) in normal Q-Gorenstein rings reduced from characteristic zero to characteristic p >> 0. Also, in fixed prime characteristic, we establish some properties of r(a) similar to those of multiplier ideals (e.g., a Briancon-Skoda-type theorem, subadditivity, etc.) with considerably simple proofs, and study the relationship between the ideal τ(α) and the F-rationality of Rees algebras.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/98ac3bde20263dfa4fe066b90d218b2a5350c2fb

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Citations in the corpus

Integral closure of ideals, rings, and modules (2006, unknown): “Yoshida [107].”
F-Thresholds, tight closure, integral closure, and multiplicity bounds (2007, background): “As we have mentioned, if R is regular and F-finite, then it was shown in [MTW] that the F-thresholds of an ideal a coincide with the jumping exponents for the generalized test ideals of [ HY ].”
IMPANGA lecture notes on log canonical thresholds (2011, background): “hold is an analogue of the log canonical threshold in positive characteristic, the test ideals give an analogue of the multiplier ideals in the same context. They have been deﬁned by Hara and Yoshida [HY] for rather general ambient varieties, and it was shown that they behave in the same way as the ”
An interpretation of multiplier ideals via tight closure (2001, methodology): “and by Matlis duality (see [ Ha3 , Lemma 3.3]), τ(R,�) ⊇ J(Y,�).”
F-signature and Hilbert–Kunz multiplicity : a combined approach and comparison (2016, unknown): “The limit s ( R ) = lim e →∞ ( 1 / p ed ) frk ( R 1 / p e ) was ﬁrst studied in [Smith and Van den Bergh 1997] and revisited in [Huneke and Leuschke 2002], where it was coined the F -signature and shown to exist for Gorenstein rings; existence in full generality was ﬁrst shown in [Tucker 2012].”
F-adjunction (2009, background): “Big test ideals are characteristic p> 0 analogs of multiplier ideals, [Smith 2000; Hara 2001; Takagi 2004b; Hara and Yoshida 2003].”
F-thresholds of hypersurfaces (2007, background): “In positive characteristic Hara and Yoshida defined in [ HY ] an analogue of the multiplier ideals, the (generalized) test ideals.”
Ordinary varieties and the comparison between multiplier ideals and test ideals (2010, methodology): “on. These ideals have recently found many striking applications in birational geometry, mostly due to their connection with vanishing theorems, see [Laz]. In positive characteristic, Hara and Yoshida [HY] introduced the notion of (generalized) test ideal, relying on a generalization of the theory of”
On a generalization of test ideals (2002, background): “ods. On the other hand, we can prove that the multiplier ideal J(at) in a normal Q-Gorenstein ring of characteristic zero coincides, after reduction to characteristic p ≫ 0, with the ideal τ(at); see [HY] and also [Ta]. In this paper, we study further properties of at-tight closure and the ideals τ(”