Ian M. Aberbach

Localization of tight closure and modules of finite phantom projective dimension

The notion of tight closure for an ideal or submodule, both for Noetherian rings of positive prime characteristic p and for finitely generated algebras over a field of characteristic 0, was introduced in [HH4]. Expository accounts are given in [HH 1]-[HH3] and [Hu 1], The theory is further explored in [HH 6], [HH 8], [HH 9], and [HH 10], äs well äs in [Ab]. In particular, [HH8] contains a detailed study of the notion of phantom homology (homology is phantom when the cycles are in the tight closure of the boundaries in the module of chains), which leads to the theory of modules of finite phantom projective dimension developed in [Ab], an important tool here. The definition of finite phantom projective dimension and the parts of the theory that we need are reviewed briefly in § 5.

A theorem of Briançon-Skoda type for regular local rings containing a field

Let (R,m) be a regular local ring containing a field. We give a refinement of the Briancon-Skoda theorem showing that if J is a minimal reduction of I where I is m-primary, then Id+w ⊆ Jw+1a where d = dimR and a is the largest ideal such that aJ = aI. The proof uses tight closure in characteristic p and reduction to characteristic p for rings containing the rationals.

Test ideals and base change problems in tight closure theory

Test ideals are an important concept in tight closure theory and their behavior via flat base change can be very difficult to understand. Our paper presents results regarding this behavior under flat maps with reasonably nice (but far from smooth) fibers. This involves analyzing, in depth, a special type of ideal of test elements, called the CS test ideal. Besides providing new results, the paper also contains extensions of a theorem by G. Lyubeznik and K. E. Smith on the completely stable test ideal and of theorems by F. Enescu and, independently, M. Hashimoto on the behavior of F-rationality under flat base change.

Finite Tor dimension and failure of coherence in absolute integral closures

It is shown both in characteristic p > 0 and in mixed characteristic p > 0 that if R is a perfect ring in the first case or RpR is perfect in the second case, then, under some additional conditions, the radical of a finitely generated ideal has finite Tor dimension, and bounds are obtained. Let R+ denote the integral closure of the domain R in an algebraic closure of its fraction field. The results are applied to show that R+ is not coherent when R is Noetherian of dimension at least 3, and, under additional restrictions, when the dimension is 2. Motivation for this question connected with tight closure theory is discussed.

Reduction numbers, Rees algebras and Pfaffian ideals

Abstract Assuming that (R, m) is a Cohen-Macaulay local ring with infinite residue field and I is an ideal of R having analytic deviation 2, we provide a condition (in terms of a presentation matrix of I, and inspired by work of Vasconcelos) that forces bounds on the reduction number of I. We proceed to apply the condition to various situations. Our main application is to a certain family of 5-generated height 3 Gorenstein ideals of a regular local ring. This application is possible by making use of the structure theorem of Buchsbaum and Eisenbud to express these Gorenstein ideals in terms of the Pfaffians of a 5 × 5 skew-symmetric presentation matrix of I. The applications help to produce Cohen-Macaulay Rees algebra results.

SOME CONDITIONS FOR THE EQUIVALENCE OF WEAK AND STRONG F-REGULARITY

ABSTRACT We show that all forms of F-regularity are equivalent for rings in which a sufficiently large symbolic power of an anti-canonical ideal has small enough analytic spread. We also show that if R is weakly F-regular and has a two-generated anti-canonical ideal then R is strongly F-regular. Both of these results then have implications for the existence of test elements.

New estimates of Hilbert–Kunz multiplicities for local rings of fixed dimension

We present results on the Watanabe–Yoshida conjecture for the Hilbert–Kunz multiplicity of a local ring of positive characteristic. By improving on a “volume estimate” giving a lower bound for Hilbert–Kunz multiplicity, we obtain the conjecture when the ring has either Hilbert–Samuel multiplicity less than or equal to 5 or dimension less than or equal to 6. For nonregular rings with fixed dimension, a new lower bound for the Hilbert–Kunz multiplicity is obtained.

Test elements in excellent rings with an application to the uniform Artin-Rees property

Working in positive prime characteristic throughout, we show that excellent rings of dimension 2 or smaller have completely stable test elements and use this to show that excellent domains of dimension 3 have the uniform Artin-Rees property.

The Briançon-Skoda theorem and coefficient ideals for non-m-primary ideals

We generalize a Brian\c{c}on-Skoda type theorem first studied by Aberbach and Huneke. With some conditions on a regular local ring

An improved Briançon-Skoda theorem with applications to the Cohen-Macaulayness of Rees algebras

(R,\m)