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Journal of the London Mathematical Society 2002 65(3):560-574; doi:10.1112/S0024610702003204
© 2002 by London Mathematical Society
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© The London Mathematical Society

Intersections of Symbolic Powers of Prime Ideals

Sean Sather-Wagstaff

Department of Mathematics, University of Illinois 273 Altgeld Hall, 1409 West Green Street, Urbana, IL 61801, USA, ssather{at}math.uiuc.edu

Received 17 April 2001. Revision received 16 October 2001.

Let (R,m) be a local ring with prime ideals p and q such that Formula. If R is regular and contains a field, and dim(R/p)+dim(R/q)=dim(R), then it is proved that p(m) {cap} q(n) {subseteq} mm+n for all positive integers m and n. This is proved using a generalization of Serre's Intersection Theorem which is applied to a hypersurface R/fR. The generalization gives conditions that guarantee that Serre's bound on the intersection dimension (R/p)+(R/q)≤dim(R) holds when R is nonregular.


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