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On a non-vanishing conjecture of Kawamata and the core of an ideal. (English) Zbl 1089.13003

Let \((A,M)\) be a local ring essentially of finite type over a field of characteristic zero with maximal ideal \(M\). For any ideal \(I\) we have the notions of integral closure of \(I\), reductions of \(I\) and the core of \(I\). The last one was introduced by D. Rees and J. D. Sally [Mich. Math. J. 35, No. 2, 241–254 (1988; Zbl 0666.13004)]. These notions are of course intimately related. More recently, yet another notion has become prominent, the adjoint ideal or the multiplier ideal of \(I\). I believe this fairly natural notion appeared in a more geometric setting in the works of Demailly. For a recent exposition see J.-P. Demailly [in: School on Vanishing Theorems and Effective Results in Algebraic Geometry, Trieste 2000, Abdus Salam Int. Cent. Theoret. Phys., ICTP Lect. Notes 6, 1–148 (2001; Zbl 1102.14300)]. The paper under review connects these concepts in a very elegant way. The authors also prove that if their results can be modified to give an appropriate graded version, then a well known conjecture of Y. Kawamata [Asian J. Math. 4, No. 1, 173–181 (2000; Zbl 1060.14505)] will follow.
For the review, let me quote just one of the results. So, let \((A,M)\) be as above and let \(I\) be \(M\)-primary. Assume that the irrelevant ideal of the Rees algebra \(A[It]\) is a Cohen-Macaulay module over \(A[It]\) and \(Y=\text{Proj}\, A[It]\) has rational singularities. Let \(\omega_A\) denote a canonical module of \(A\). Then \[ \text{core}\,(I\omega_A)=\text{adj}\,(I^d\omega_A):= \Gamma(Y,I^d\omega_Y). \] More results in this direction can also be found in the authors’ article: [E. Hyry and K. E. Smith, Trans. Am. Math. Soc. 356, No. 8, 3143–3166 (2004; see the following review Zbl 1089.13004)].

MSC:

13A15 Ideals and multiplicative ideal theory in commutative rings
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13D45 Local cohomology and commutative rings
14C20 Divisors, linear systems, invertible sheaves
14F17 Vanishing theorems in algebraic geometry
13B22 Integral closure of commutative rings and ideals
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