Graded annihilators and tight closure test ideals

Geometric Interpretation of Tight Closure and Test Ideals

We study tight closure and test ideals in rings of characteristic p 0 using resolution of singularities. The notions of F -rational and F regular rings are defined via tight closure, and they are known to correspond with rational and log terminal singularities, respectively. In this paper, we reformulate this correspondence by means of the notion of the test ideal, and generalize it to wider cl...

Integral closure of ideals and annihilators of homology

Tight Closure in Graded Rings

This paper facilitates the computation of tight closure by giving giving upper and lower bounds on the degrees of elements that need to be checked for inclusion in the tight closure of certain homogeneous ideals in a graded ring. Differential operators are introduced to the study of tight closure, and used to prove that the degree of any element in the tight closure of a homogeneous ideal (but ...

A Generalization of Tight Closure and Multiplier Ideals

We introduce a new variant of tight closure associated to any fixed ideal a, which we call a-tight closure, and study various properties thereof. In our theory, the annihilator ideal τ(a) of all a-tight closure relations, which is a generalization of the test ideal in the usual tight closure theory, plays a particularly important role. We prove the correspondence of the ideal τ(a) and the multi...

Localization and Test Exponents for Tight Closure

We introduce the notion of a test exponent for tight closure, and explore its relationship with the problem of showing that tight closure commutes with localization, a longstanding open question. Roughly speaking, test exponents exist if and only if tight closure commutes with localization: mild conditions on the ring are needed to prove this. We give other, independent, conditions that are nec...