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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A generalization of tight closure and multiplier ideals

Authors: Nobuo Hara and Ken-ichi Yoshida
Journal: Trans. Amer. Math. Soc. 355 (2003), 3143-3174
MSC (2000): Primary 13A35, 14B05
Published electronically: April 11, 2003
MathSciNet review: 1974679
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Abstract: We introduce a new variant of tight closure associated to any fixed ideal $\mathfrak{a}$, which we call $\mathfrak{a}$-tight closure, and study various properties thereof. In our theory, the annihilator ideal $\tau (\mathfrak{a})$ of all $\mathfrak{a}$-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 $\tau (\mathfrak{a})$ and the multiplier ideal associated to $\mathfrak{a}$ (or, the adjoint of $\mathfrak{a}$ in Lipman's sense) in normal $\mathbb{Q} $-Gorenstein rings reduced from characteristic zero to characteristic $p \gg 0$. Also, in fixed prime characteristic, we establish some properties of $\tau (\mathfrak{a})$ similar to those of multiplier ideals (e.g., a Briançon-Skoda-type theorem, subadditivity, etc.) with considerably simple proofs, and study the relationship between the ideal $\tau (\mathfrak{a})$ and the F-rationality of Rees algebras.

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Additional Information

Nobuo Hara
Affiliation: Mathematical Institute, Tohoku University, Sendai 980–8578, Japan

Ken-ichi Yoshida
Affiliation: Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464–8602, Japan

Received by editor(s): August 20, 2002
Received by editor(s) in revised form: December 19, 2002
Published electronically: April 11, 2003
Additional Notes: Both authors are partially supported by a Grant-in-Aid for Scientific Research, Japan
Article copyright: © Copyright 2003 American Mathematical Society