On symbolic powers of prime ideals

Details On symbolic powers of prime ideals On symbolic powers of prime ideals

2002-07-08

arxiv.org/abs/math/0207068v2

Mathematics

Commutative Algebra

14 pages; uses XY-pic and conm-p-l.cls

Scientific paper

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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