An interpretation of multiplier ideals via tight closure

By S. Takagi. 2001

116 citations recorded by the source corpus.

Hara [Ha3] and Smith [Sm2] independently proved that in a normal Q-Gorenstein ring of characteristic p 0, the test ideal coincides with the multiplier ideal associated to the trivial divisor. We extend this result for a pair (R,∆) of a normal ring R and an effective Q-Weil divisor ∆ on SpecR. As a corollary, we obtain the equivalence of strongly F-regular pairs and klt pairs.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/4aaf8019f27cfc784a103e21c9af9846bddc8f02

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Citations in the corpus

Test ideals in non-Q-Gorenstein rings (2009, methodology): “On the other hand, Hara, Yoshida and Takagi deﬁned the big test ideal τ b ( R ; Δ , a t ) of a triple ( X, Δ , a t ) and showed it agreed with the multiplier ideal J ( X ; Δ , a t ) after reduction to characteristic p 0; see [HY03] and [Tak04b].”
F-signature and Hilbert–Kunz multiplicity : a combined approach and comparison (2016, unknown): “Hara and Watanabe [2002] have deﬁned tight closure for divisor pairs (compare to [Takagi 2004]), and Hara and Yoshida [2003] have done the same for ideals with a real coefﬁcient; both of these works build on the important results of Smith [1997], Hara [1998], and Mehta and Srinivas [1997].”
Singularities on normal varieties (2008, background): “In the (log) Q-Gorenstein setting, multiplier ideals reduce, for sufficiently large characteristics, to the corresponding generalized test ideals (see [Smi, Har, HY, Tkg ]).”
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-singularities via alterations (2011, background): “ment that the test ideal is contained in the multiplier ideal (since the test ideal is the uniquesmallest ideal satisfying a certain property), that Trg(g∗ωY (−⌊f∗(KX +∆)⌋)) ⊇τ(Z;−KZ +h∗(KX +∆)). See [Tak04] or see [ST12a, Theorem 4.17] for a sketch of a simpler version of this argument (or see imme”
Integral closure of ideals, rings, and modules (2006, background): “It has been further studied in Hara [105], [106]; Smith [275]; Takagi [287]; Hara and Takagi [107]; and Hara and Watanabe [109].”
Globally F-regular and log Fano varieties (2009, background): “ng. Does there exist an eective Q-divisor on Spec Rsuch that K R+ is Q-Cartier and e˝(R) = ˝e(R;)? For the denition of ~˝we refer to the paper [LS01] for the original boundary-less denition and to [Tak04b] for the general case, also see [HT04]. The following question is natural in view of Lemma 4.9 ”
Discreteness and rationality of F-thresholds (2006, methodology): “In this section we collect some basic results about generalized test ideals as introduced by Hara and Yoshida [HY] (see also [ Tak ] and [HT]).”
Centers of F-purity (2008, methodology): “We now show that big sharp test elements in this generality always exist, see [19, Theorem 5.9], [13, 40 ].”
On a generalization of test ideals (2002, background): “On the other hand, we can prove that the multiplier ideal J(a t ) in a normal Q-Gorenstein ring of characteristic zero coincides, after reduction to characteristic p ≫ 0, with the ideal τ(a t ); see [HY] and also [ Ta ].”
Discreteness and rationality of F-jumping numbers on singular varieties (2009, background): “ays (see [MY09]). Test ideals are rooted in the theory of tight closure of Hochster and Huneke [HH90], see also [Smi95] and [Har01]. The variants we are considering here were introduced in [HY03] and [Tak04]. Just as for multiplier ideals, as one increases t, the test ideal τ(X,at) becomes a smaller”