Test Elements, Analogues of Tight Closure, and Size for Ideals

Abstract

We give many new results related to the theory of tight closure and its generalizations. Explicitly, we establish a series of results showing that the Jacobian ideal is contained in the test ideal for tight closures both in equal characteristic p and equal characteristic 0 for algebras essentially of finite type over power series rings (they are called semianalytic algebras). We move on to introduce and study a new closure called wepf in mixed characteristic, and prove that it is a Dietz closure satisfying the Algebra axiom. This is the first known example of a Dietz closure in mixed characteristic. This is achieved by proving that the epf closure satisfies what we call the p-colon-capturing property. We define and study the relationships with properties connected with tight closure. For example, we show that a persistent closure operation that captures colons automatically captures the plus closure, i.e., the contraction of the expansion of an ideal to the absolute integral closure of the ring. We also show that the existence of persistent closure operations between two complete local domains gives us a weakly functorial version of the existence of big Cohen-Macaulay algebras for them. We also develop a new numerical notion for ideals called size using the theory of quasilength, and show that the size of an ideal is always between its height and arithmetic rank. We show under mild conditions that the size is the same as height for one-dimensional primes in a local ring whose completion is a domain. We further study the additive property and the asymptotic additive property of quasilength.