The Blowup Closure of a Set of Ideals with Applications to TI Closure

Abstract

We introduce a new closure operation on sets of ideals in a commutative Noetherian ring of characteristic p, called the blowup closure. We develop the theory of this operation and prove that given a set of ideals in a Noetherian ring of characteristic p, under mild conditions, the tight integral closure of this set agrees with the blowup closure of certain extensions of those ideals in an extension of the original ring. We then use properties of blowup closure to settle open questions on tight integral closure posed by M. Hochster (J. Algebra230 (2000), 184–203). In particular, we show that under mild conditions on the ring, tight integral closure persists under ring maps and that it commutes with localization if and only if tight closure does.