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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A generalization of tight closure and multiplier ideals
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by Nobuo Hara and Ken-ichi Yoshida PDF
Trans. Amer. Math. Soc. 355 (2003), 3143-3174 Request permission

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
  • Email: hara@math.tohoku.ac.jp
  • Ken-ichi Yoshida
  • Affiliation: Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464–8602, Japan
  • MR Author ID: 359418
  • Email: yoshida@math.nagoya-u.ac.jp
  • 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
  • © Copyright 2003 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 355 (2003), 3143-3174
  • MSC (2000): Primary 13A35, 14B05
  • DOI: https://doi.org/10.1090/S0002-9947-03-03285-9
  • MathSciNet review: 1974679