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On symbolic powers of prime ideals

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Sean Sather-Wagstaff

Commutative Algebra · Submitted 8 Jul 2002

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In particular, we prove that this conjecture holds for all regular local rings if and only if it holds for all localizations of polynomial algebras over complete discrete valuation rings.

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arXiv IDmath/0207068 (version 2)
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CitationSean Sather-Wagstaff. "On symbolic powers of prime ideals." arXiv:math/0207068 (2002). https://pith.science/paper/AHTHQUDE
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,230 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.47 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.35 · compares with prior results 0.48 · mentions code or data availability 0.01.

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arXiv listing abstract v2. Results and the main claim are underlined; .

Let (R,m) be a regular local ring with prime ideals p and q such that p+q is m-primary and dim(R/p)+dim(R/q)=dim(R). It has been conjectured by Kurano and Roberts that p^{(n)} \cap q \subseteq m^{n+1} for all positive integers n. We discuss this conjecture and related conjectures. In particular, we prove that this conjecture holds for all regular local rings if and only if it holds for all localizations of polynomial algebras over complete discrete valuation rings. In addition, we give examples showing that certain generalizations to nonregular rings do not hold.

background contribution method result interpretation main claim

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 Let (R,m) be a regular local ring with prime ideals p and q such that p+q is m-primary and dim(R/p)+dim(R/q)=dim(R). background 1.00 0.10 0.02 asserted 0.78
s1 It has been conjectured by Kurano and Roberts that p^{(n)} \cap q \subseteq m^{n+1} for all positive integers n. background 1.00 0.03 0.08 speculative 0.95
s2 We discuss this conjecture and related conjectures. contribution 0.93 0.81 0.03 unclear 0.64
s3 In particular, we prove that this conjecture holds for all regular local rings if and only if it holds for all localizations of polynomial algebras over complete discrete valuation rings. result 0.96 0.92 0.70 asserted 1.00
s4 In addition, we give examples showing that certain generalizations to nonregular rings do not hold. result 0.90 0.95 0.06 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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The paper
Original abstract

Let (R,m) be a regular local ring with prime ideals p and q such that p+q is m-primary and dim(R/p)+dim(R/q)=dim(R). It has been conjectured by Kurano and Roberts that p^{(n)} \cap q \subseteq m^{n+1} for all positive integers n. We discuss this conjecture and related conjectures. In particular, we prove that this conjecture holds for all regular local rings if and only if it holds for all localizations of polynomial algebras over complete discrete valuation rings. In addition, we give examples showing that certain generalizations to nonregular rings do not hold.

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