Premium
Intersections of Symbolic Powers of Prime Ideals
Author(s) -
Sather-Wagstaff Keri Ann
Publication year - 2002
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610702003204
Subject(s) - mathematics , hypersurface , prime (order theory) , generalization , combinatorics , intersection (aeronautics) , complete intersection , dimension (graph theory) , ring (chemistry) , field (mathematics) , discrete mathematics , ring of integers , pure mathematics , algebraic number field , mathematical analysis , chemistry , organic chemistry , engineering , aerospace engineering
Let ( R ,m) be a local ring with prime ideals p and q such thatp + q = m . If R is regular and contains a field, and dim( R /p)+dim( R /q)=dim( R ), then it is proved that p ( m ) ∩ q ( n ) ⊆ m m + 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.