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Test ideals vs. multiplier ideals. (English) Zbl 1162.13004

In this interesting paper, the authors compare the behavior of the test ideal to the behavior of the multiplier ideal. The test ideal \(\tau(f^t)\), which comes out of tight closure theory, is closely related to the multiplier ideal \(\mathcal{J}(f^t)\), which is defined by resolution of singularities in characteristic zero. See the paper by N. Hara and K.-i. Yoshida [Trans. Am. Math. Soc. 355, No. 8, 3143–3174 (2003; Zbl 1028.13003)] for background and definitions.
It turns out that many interesting and difficult-to-prove theorems about multiplier ideals have simple proofs for test ideals. However, as this paper shows, many basic facts about multiplier ideals fail spectacularly for test ideals. In particular, because multiplier ideals are defined by resolution of singularities, they are always integrally closed (furthermore, the local syzygies satisfy certain conditions, see the paper of R. Lazarsfeld and K. Lee [Invent. Math. 167, No. 2, 409–418 (2007; Zbl 1114.13013)]). However, the main result of this paper, stated below, shows that every ideal (including non-integrally closed ideals) is a test ideal.
Theorem 1.1. Suppose that \(R\) is an \(F\)-finite regular local ring of characteristic \(p > 0\). For every ideal \(I\) in \(R\), there is \(f \in R\) and \(c > 0\) such that \(I = \tau(f^c)\).
The authors also show several other common properties of multiplier ideals completely fail for test ideals (for example, \(\tau(f^c)\) need not be a radical ideal when \(c\) is the \(F\)-pure threshold). See the paper for details.
As the authors point out, most of the pathological behavior in characteristic \(p > 0\) occurs for test ideals \(\tau(f^c)\) when \(c\) is a rational number with \(p\) in the denominator. It may be that when the \(F\)-jumping numbers of \(\tau(f^c)\) do not have \(p\) in the denominator, the test ideal behaves more like the multiplier ideal.

MSC:

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

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