Test Ideals in Diagonal Hypersurface Rings

McDermott, Moira A.

doi:10.1006/jabr.2000.8573

Cited by 7 publications

(10 citation statements)

References 4 publications

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“…, x n ) d−n+1 for sufficiently large p, and if d < n, then the test ideal is the unit ideal [Hu2,6.3]. Fedder and Watanabe also have results that show that if d < n, then the ring is F-regular [FW,(2.11)] and hence the test ideal is the unit ideal, again for sufficiently large p. On the other hand, we show in [McD2] that if p < d, then the test ideal is contained in (x 1 , . .…”

Section: Introductionsupporting

confidence: 52%

“…In many examples when the dimension is greater than two, the bound of p > d is sufficient. For this reason, and the fact that the case when p = d − 1 is known [McD2], we are particularly interested in computing test ideals when p < d − 1. We do have one example (4.3) where p is greater than d but less than the bound in [FW] (p > n(d − 1) − d), and the ring is not F -regular as predicted.…”

Section: Introductionmentioning

confidence: 99%

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Test ideals in diagonal hypersurface rings, II

McDermott

2003

Journal of Algebra

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Let R = k[x 1 , . . . , x n ]/(x d 1 + · · · + x d n ), where k is a field of characteristic p, p does not divide d and n ≥ 3. We describe a method for computing the test ideal for these diagonal hypersurface rings. This method involves using a characterization of test ideals in Gorenstein rings as well as developing a way to compute tight closures of certain ideals despite the lack of a general algorithm. In addition, we compute examples of test ideals in diagonal hypersurface rings of small characteristic (relative to d) including several that are not integrally closed. These examples provide a negative answer to Smith's question of whether the test ideal in general is always integrally closed [Sm2].

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Section: Introductionsupporting

confidence: 52%

Section: Introductionmentioning

confidence: 99%

Test ideals in diagonal hypersurface rings, II

McDermott

2003

Journal of Algebra

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“…Note that test ideals of diagonal hypersurfaces, in the sense of Hochster and Huneke, were computed by McDermott in [McD01,McD03]. Motivated by the connections mentioned above, we are especially interested understanding the behavior of these invariants for all (or for all but finitely many) possible values of p.…”

Section: Introductionmentioning

confidence: 99%

$F$-invariants of diagonal hypersurfaces

Hernández

2014

Proc. Amer. Math. Soc.

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In this note, we derive formulas for the F -pure threshold, higher jumping numbers, and test ideals of diagonal and Fermat hypersurfaces. For these hypersurfaces, we answer a question of Schwede regarding the denominators of F -pure thresholds, and obtain tight upper bounds for the number of higher jumping numbers. Our results are valid over all (or all but finitely many) characteristics, and therefore allow us to construct examples in which the characteristic p setting is drastically different than that over C.

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“…For example, it follows immediately from the definition that every multiplier ideal is integrally closed (we suggest the reader prove this as an exercise). However, not every test ideal is integrally closed [McD03] and furthermore, every ideal in a regular ring is the test ideal of an appropriate pair τ (R, f t ), see [MY09] (here (R, f t ) is a pair as discussed in Section 6 below).…”

Section: 3mentioning

confidence: 99%

A Survey of Test Ideals

2012

Progress in Commutative Algebra 2

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Introduction 1 2. Characteristic p preliminaries 3 3. The test ideal 5 4. Connections with algebraic geometry 9 5. Tight closure and applications of test ideals 15 6. Test ideals for pairs (R, a t ) and applications 19 7. Generalizations of pairs: algebras of maps 23 8. Other measures of singularities in characteristic p 25 Appendix A. Canonical modules and duality 31 Appendix B. Divisors 34 Appendix C. Glossary and diagrams on types of singularities 35 References 38 . . . xIf M is any R-module, the (geometrically motivated) notation F e * M is often used to denote the corresponding R-module coming from restriction of scalars for F e . Thus, M and F e * M agree as both sets and Abelian groups. However, if F e * m denotes the element of F e * M corresponding to m ∈ M , we have r • F e * m = F e * (r p e • m) for r ∈ R and m ∈ M . It is easy to see that F e * R and R 1/p e are isomorphic R-modules by identifying F e * r with r 1/p e for each r ∈ R. While we have taken preference to the use of R 1/p e throughout, it can be very helpful to keep both perspectives in mind.Remark 2.2. We caution the reader that the module F e * M is quite different from that which is commonly denoted F e (M ) originating in [PS73]. This latter notation coincides rather with M ⊗ R F e * R considered as an R-F e * R bimodule.Exercise 2.3. Show that F e * ( ) is an exact functor on the category of R-modules. Conclude that F e * (R/I) and R 1/p e /I 1/p e are isomorphic R-modules (and (F e * R = R 1/p e )-modules) for any ideal I ⊆ R.Lemma 2.4. R 1/p e is a finitely generated R-module.1 Essentially all of the positive characteristic material in this paper can easily be generalized to the setting of reduced F -finite rings. In addition, large portions of the theory extend to the setting of excellent local rings.

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Test Ideals in Diagonal Hypersurface Rings

Cited by 7 publications

References 4 publications

Test ideals in diagonal hypersurface rings, II

Test ideals in diagonal hypersurface rings, II

$F$-invariants of diagonal hypersurfaces

A Survey of Test Ideals

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