Comparison of symbolic and ordinary powers of ideals

Details Comparison of symbolic and ordinary powers of ideals Comparison of symbolic and ordinary powers of ideals

2002-11-11

arxiv.org/abs/math/0211174v1

Invent. Math. 147 (2002), no. 2, 349--369

Mathematics

Commutative Algebra

Scientific paper

10.1007/s002220100176

In this paper we generalize the theorem of Ein-Lazarsfeld-Smith (concerning the behavior of symbolic powers of prime ideals in regular rings finitely generated over a field of characteristic 0) to arbitrary regular rings containing a field. The basic theorem states that in such rings, if P is a prime ideal of height c, then for all n, the symbolic (cn)th power of P is contained in the nth power of P. Results are also given in the non-regular case: one must correct by a power of the Jacobian ideal in rings where the Jacobian ideal is defined.

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