Uniform equivalence of symbolic and adic topologies

Huneke, Craig; Katz, Daniel J.; Validashti, Javid

doi:10.1215/ijm/1264170853

Cited by 22 publications

(20 citation statements)

References 22 publications

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“…The following proposition is implicit in the work of McAdam [McA87] and Schenzel [Sch86]. We present the proof given by Huneke, Katz and Validashti [HKV09]. Proof.…”

Section: Linear Equivalence Of Topologiesmentioning

confidence: 93%

“…Theorem 3.17. [Huneke-Katz-Validashti [HKV09]] Let R be an equicharacteristic local domain such that R is an isolated singularity. Assume that R is either essentially of finite type over a field of characteristic zero or R has positive characteristic and is F -finite.…”

Section: Linear Equivalence Of Topologiesmentioning

confidence: 99%

“…We now focus on a result needed to prove uniform bounds for isolated singularities. This result comes from the work of Hochster and Huneke [HH02], where the following property of the Jacobian ideal plays a crucial role in the main results concerning uniform linearity of symbolic powers over regular local rings: Moreover, a parallel result is proved for F -finite local rings in characteristic p with isolated singularity [HKV09]. In this case, J can be chosen to be a fixed m-primary ideal, though not necessarily the square of the Jacobian ideal.…”

Section: Linear Equivalence Of Topologiesmentioning

confidence: 99%

See 2 more Smart Citations

Symbolic Powers of Ideals

Dao

Stefani²,

Grifo

et al. 2018

Springer Proceedings in Mathematics &Amp; Statistics

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“…The following proposition is implicit in the work of McAdam [McA87] and Schenzel [Sch86]. We present the proof given by Huneke, Katz and Validashti [HKV09]. Proof.…”

Section: Linear Equivalence Of Topologiesmentioning

confidence: 93%

Section: Linear Equivalence Of Topologiesmentioning

confidence: 99%

Section: Linear Equivalence Of Topologiesmentioning

confidence: 99%

See 1 more Smart Citation

Symbolic Powers of Ideals

Dao

Stefani²,

Grifo

et al. 2018

Springer Proceedings in Mathematics &Amp; Statistics

Self Cite

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“…Let us mention that it is known that if (R, m) is a regular local ring of dimension d, then I (dc) ⊂ I c for any ideal I of R and any integer c [4]. If (R, m) is an isolated singularity ring, then I (kc) ⊂ I c for any ideal I of R and any integer c [5] for some constant k independent of p. For a general local ring R and for any ideal I, there exists a constant k depending on I such that I (kc) ⊂ I c for any c [16] but it is still an open question whether such a k may be chosen independently of I in general.…”

Section: Applicationsmentioning

confidence: 99%

The analogue of Izumi's Theorem for Abhyankar valuations

Rond¹,

Spivakovsky²

2014

Journal of the London Mathematical Society

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Abstract. A well known theorem of Shuzo Izumi, strengthened by David Rees, asserts that all the divisorial valuations centered in an analytically irreducible local noetherian ring (R, m) are linearly comparable to each other. This is equivalent to saying that any divisorial valuation ν centered in R is linearly comparable to the m-adic order. In the present paper we generalize this theorem to the case of Abhyankar valuations ν with archimedian value semigroup Φ. Indeed, we prove that in a certain sense linear equivalence of topologies characterizes Abhyankar valuations with archimedian semigroups, centered in analytically irreducible local noetherian rings. In other words, saying that R is analytically irreducible, ν is Abhyankar and Φ is archimedian is equivalent to linear equivalence of topologies plus another condition called weak noetherianity of the graded algebra gr ν R.We give some applications of Izumi's theorem and of Lemma 2.7, which is a crucial step in our proof of the main theorem. We show that some of the classical results on equivalence of topologies in noetherian rings can be strengthened to include linear equivalence of topologies. We also prove a new comparison result between the m-adic topology and the topology defined by the symbolic powers of an arbitrary ideal.

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“…Since then, it has been of great interest to know which non-regular rings have USTP. For instance, Huneke-Katz-Validashti showed that, under suitable hypotheses, rings with isolated singularities have USTP, although without an effective bound on h [HKV09]. R. Walker showed that 2-dimensional rational singularities have USTP and obtained an effective bound for h [Wal16].…”

Section: Introductionmentioning

confidence: 99%

The uniform symbolic topology property for diagonally F-regular algebras

Carvajal-Rojas

Smolkin

2020

Journal of Algebra

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Let k be a field of positive characteristic. Building on the work of [Smo18], we define a new class of k-algebras, called diagonally F -regular algebras, for which the so-called Uniform Symbolic Topology Property (USTP) holds effectively. We show that this class contains all essentially smooth k-algebras. We also show that this class contains certain singular algebras, such as the affine cone over P r k × P s k , when k is perfect. By reduction to positive characteristic, it follows that USTP holds effectively for the affine cone over P r C × P s C and more generally for complex varieties of diagonal F -regular type.holds for our choice of t. Following [Smo18], we will construct t using the so-called diagonal Cartier algebras. Namely, we setwhere D (n) is the n-th diagonal Cartier algebra; see Definition 3.1. Then Proposition 3.4(c) demonstrates that containment (2 ′ ) holds for any reduced k-algebra essentially of finite type, while (1 ′ ) holds whenever D (n) is F -regular. When this is the case for all n, we say our ring is diagonally F -regular as a k-algebra. This sketches the proof of our main theorem:1 If k is perfect, then essentially smooth k-algebras are the same as regular k-algebras essentially of finite type. 2 Also known as "Cartesian products," as in [Har77, Ch. II, Exc. 5.11]. 3 Ein-Lazarsfeld-Smith's original argument uses multiplier ideals, which are only known to exist in characteristic 0. Their argument was adapted to positive characteristic rings by N. Hara [Har05] and to mixedcharacteristic rings by L. Ma and K. Schwede [MS17]. Hara and Ma-Schwede achieved this by using positive characteristic and mixed characteristic analogs of multiplier ideals, respectively. * R, R). More generally, given a finite R-module M we may define a Cartier algebra C M over R as R in degree zero and as C e,M := Hom R (F e * M, M) in higher degrees. The ring multiplication of C M is defined by the rule ϕ e · ϕ d := ϕ e • F e * ϕ d for all ϕ e ∈ C e,M , ϕ d ∈ C d,M . Furthermore, the left R-module structure of C M is the usual one given by post-multiplication,whereas the right R-module structure is given by pre-multiplication by elements of F e * R. More precisely, if ϕ ∈ C e,M and r ∈ R, then (ϕ · r)(−) = ϕ(F e * r · −) It is worth mentioning we are primarily concerned with Cartier subalgebras of C R in this work.Definition 2.2 (Cartier Modules). Given a ring R and a Cartier R-algebra C, we define a Cartier C-module to be a finite R-module M equipped with a homomorphism C − → C M of Cartier R-algebras. This is the same as saying M is a left C-module with coherent underlying R-module structure [BS16, Lemma 5.2]. A morphism of Cartier C-modules is defined to be a morphism of left C-modules.Let R be a ring and C a Cartier R-algebra. Under the assumption R is essentially of finite type over k, Blickle and Stäbler constructed a covariant functor τ = τ(−, C) : left-C-mod − → R-mod, from the category of Cartier C-modules to the category of R-modules. This functor is an additive subfunctor of the forgetful functor between ...

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Uniform equivalence of symbolic and adic topologies

Cited by 22 publications

References 22 publications

Symbolic Powers of Ideals

Symbolic Powers of Ideals

The analogue of Izumi's Theorem for Abhyankar valuations

The uniform symbolic topology property for diagonally F-regular algebras

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