Tight closure commutes with localization in binomial rings

Abstract

It is proved that tight closure commutes with localization in any domain which has a module finite extension in which tight closure is known to commute with localization. It follows that tight closure commutes with localization in rings, in particular in semigroup or toric rings. Tight closure, introduced in [HHI], is a closure operation performed on ideals in a commutative, Noetherian ring containing a field. Although it has many important applications to commutative algebra and related areas, some basic questions about tight closure remain open. One of the biggest such open problems is whether or not tight closure commutes with localization. For an exposition of tight closure, including the localization problem, see [Hu]. The purpose of this paper is give a simple proof that tight closure commutes with localization for a class of rings that includes semi-group rings and, slightly more generally, the so-called binomial rings, that is, those that are a quotient of a finitely generated algebra by an ideal generated by binomials. Although this is a modest class of rings, the result here includes coordinate rings of all affine toric varieties, and is general enough to substantially improve previous results of a number a mathematicians, including Moty Katzman, Will Traves, Keith Pardue and Karen Chandler, Irena Swanson and myself. Throughout this note, R denotes a Noetherian commutative ring of prime characteristic. We say that tight closure commutes with localization in R if, for all ideals I in R and all multiplicative systems U in R, I*R[U-1] = (IR[U-1])*. The main result is the following: Theorem. Let R be a ring with the following property: for each minimal prime P of R, the quotient R/P has an integral extension domain in which tight closure commutes with localization. Then tight closure commutes with localization in R. Lemma 1. If tight closure commutes with localization in R/P for each minimal prime P of R, then tight closure commutes with localization in R. Received by the editors January 11, 1999 and, in revised form, May 15, 1999. 1991 Mathematics Subject Classification. Primary 13A35.