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A generalization of tight closure and multiplier ideals. (English) Zbl 1028.13003

The authors introduce a new variant of tight closure: For any fixed ideal \(a\) in a ring \(R\) of positive prime characteristic \(p\), the \(a\)-tight closure \(I^{sa}\) is the set of all \(x \in R\) for which there exists \(c \in R\) not in any minimal prime ideal, such that for all \(q = p^e\), \(c x^q a^q \subseteq I^{[q]}\), where \(I^{[q]}\) is the \(e\)th Frobenius power of \(I\). In case \(a = R\), this is the definition of the usual tight closure of \(I\), as defined by M. Hochster and C. Huneke [J. Am. Math. Soc. 3, 31-116 (1990; Zbl 0701.13002)]. The paper under review develops the theory of \(a\)-tight closure, and is rich in theory and applications. Hara and Yoshida prove that the \(a\)-tight closure is not a true closure operation, namely that \(I^{sa}\) in general need not equal \((I^{sa})^{sa}\), nevertheless, they prove that it is still very useful. In section 2 they prove that a criterion from a paper by N. Hara, K.-I. Watanabe and K. Yoshida [J. Algebra 247, 153-190 (2002; Zbl 1008.13001)] of F-rationality of a Rees algebra can be expressed using \(a\)-tight closure. Also, under the usual assumptions of the tight closure theory, the authors prove that \(a\)-test elements exist. Furthermore, the ideals \(\tau(a)\) of \(a\)-test elements satisfy several interesting properties, including a form of the Briançon-Skoda theorem [see J. Lipman, Math. Res. Lett. 1, 739-755 (1994; Zbl 0844.13015) for a form using adjoint ideals]. In sections 3 and 4 the authors prove some connections between \(a\)-test ideals and multiplier ideals. Some of such connections had been proved by the author and others [N. Hara, Trans. Am. Math. Soc. 35, 1885-1906 (2001; Zbl 0976.13003), K. E. Smith, Commun. Algebra 28, 5915-5929 (2000; Zbl 0979.13007) and S. Takagi, An interpretation of multiplier ideals via tight closure (preprint)]. In the paper under review, theorem 4.5 asserts subadditivity of test ideals: \(\tau(ab) \subseteq \tau(a) \tau(b)\). Compare this with subadditivity of multiplier ideals in the paper by J.-P. Demailly, L. Ein and R. Lazarsfeld [Mich. Math. J. 48, Spec. Vol., 137-156 (2000; Zbl 1077.14516)]. Theorem 3.4 more generally asserts that in a sense, the characteristic \(p\) proofs of \(a\)-test ideals prove the properties of multiplier ideals in characteristic \(0\). Section 5 was motivated by E. Hyry’s papers [Manuscr. Math. 98, 377-390 (1999; Zbl 0933.13006) and Proc. Am. Math. Soc. 129, 1299-1308 (2001; Zbl 0986.13003)]: It shows connections between F-rationality of the Rees algebra of \(I\) and test ideals of powers of \(I\). In the last section the authors generalize \(a\)-test ideals to \(a\)-test ideals with rational coefficients. The motivation for this was the corresponding theory of multiplier ideals with rational coefficients, as in the paper by L. Ein, R. Lazarsfeld and K. E. Smith [Invent. Math. 144, 241-252 (2001; Zbl 1076.13501) and R. Lazarsfeld, “Positivity in algebraic geometry” (preprint)]. More on this and another form of the Briançon-Skoda theorem can be found in the subsequent paper by N. Hara and S. Takagi, “Some remarks on generalization of test ideals” (preprint).

MSC:

13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
14B05 Singularities in algebraic geometry
13B22 Integral closure of commutative rings and ideals
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References:

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