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.


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


[1] DOI: 10.2307/2373351 · Zbl 0188.33702
[2] Positivity in Algebraic Geometry II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, A Series of Modern Surveys in Mathematics 49 (2004)
[3] DOI: 10.1090/S0002-9947-03-03285-9 · Zbl 1028.13003
[4] Nagoya Math. J. 175 pp 59– (2004) · Zbl 1094.13004
[5] DOI: 10.4310/MRL.2006.v13.n5.a6 · Zbl 1110.14005
[6] DOI: 10.1090/S0894-0347-05-00481-9 · Zbl 1075.14001
[7] DOI: 10.1016/j.jalgebra.2004.07.011 · Zbl 1082.13004
[8] DOI: 10.1215/S0012-7094-04-12333-4 · Zbl 1061.14003
[9] DOI: 10.1353/ajm.2006.0049 · Zbl 1109.14005
[10] F-thresholds of hypersurfaces, Trans. Amer. Math. Soc., to appear · Zbl 1193.13003
[11] DOI: 10.4310/MRL.2008.v15.n6.a14 · Zbl 1185.13010
[12] DOI: 10.1307/mmj/1220879396 · Zbl 1177.13013
[13] Eur. Math. Soc., Zürich pp 341– (2005)
[14] DOI: 10.4310/MRL.2003.v10.n4.a2 · Zbl 1049.13003
[15] DOI: 10.1007/s00222-006-0019-9 · Zbl 1114.13013
[16] DOI: 10.1006/jabr.2001.8998 · Zbl 1008.13001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.