Generalized Test Ideals and Symbolic Powers

摘要

Ein, Lazarsfeld, and Smith proved in [ELS] the following uniform behavior of symbolic powers of ideals in affine regular rings of equal characteristic 0: If h is the largest height of any associate prime of an ideal I is contained in R, then I~((hn+kn)) is contained in (I~((k+1)))~n for all integers n ≥ 1 and k ≥ 0. Here, if W is the complement of the union of the associate primes of I, then the mth symbolic power I~((m)) of I is defined to be the contraction of I~mR_W to R, where R_W is the localization of R at the multiplicative system W.