Commutative Algebra: Homological and Birational Theory

In this part of the proposal we outline the topics listed above and explain in more detail the current state of research and why a workshop in this area would be very useful at the present time.

1. Problems in positive and mixed characteristic.

For rings of positive characteristic, the Frobenius map, which sends an element of the ring to its pth power, where p is the characteristic of the ring, is an extremely powerful tool. Many conjectures on the homological theory of Noetherian rings have been proven using this technique, and more recently these ideas have been extended in the theory of tight closure. For rings of mixed characteristic, an important line of research has been to prove these conjectures and to extend these ideas to that case.

Positive characteristic and tight closure.

The theory of tight closure was introduced by Hochster and Huneke and is currently a very active area. Connections have been proven to exist between the concepts which arise naturally in this theory and classical properties of singularities by Karen Smith, Watanabe, and others, and there is now a considerable amount of interest in relations with deep properties of local cohomology. Among the current topics of research in this area are test ideals, preservation under base change, and relations with local cohomology, including work by Smith, Lyubeznik, and Singh. Much of this work is related to one of the fundamental questions, whether tight closure commutes with localization. One of the new developments which we intend to include in our workshop is the connection with mutliplier ideals, a topic we discuss in more detail in the third section.

Mixed characteristic and the homological conjectures.

During the last year a major step forward was made in the homological conjectures for rings of mixed characteristic with Heitmann's proof of the Direct Summand Conjecture in dimension three. This case of the conjecture had been open and had been the subject of intense investigations for over thirty years. In addition to solving an important problem, the ideas used in the proof are related to methods using the Frobenius map and tight closure which have been shown to be very useful in studying problems in the case of equal characteristic. While Heitmann's results do not quite extend this theory to the mixed characteristic case, they do show that many of the ideas carry over and can be used successfully.

The new methods also have counterparts from an unexpected source, that of Arithmetic Geometry. The results outlined in the previous section have also been studied by Faltings, Gabber, Ramero, and others in their work on almost etale extensions with different aims but with some very similar results, including a result on the vanishing of local cohomology similar to that of Heitmann. One of the aims of this workshop is to invite mathematicians from these different areas to combine the expertise of these groups working on related questions.

Hilbert-Kunz multiplicities.

The third topic is Hilbert-Kunz functions and multiplicities. These invariants are an analogue of traditional multiplicities in Algebraic Geometry and record the action of the Frobenius homomorphism on a ring in positive characteristic. They are related to local Chern classes and other arithmetic invariants. Recent results by Fakruddin and Trevedi use vanishing theorems by Haboush and Andersen to study cones over elliptic curves in positive characteristic. In addition, Monsky and his student Teixeira have found an exciting description in terms of dynamical systems on curves over finite fields that yields rationality of the Hilbert-Kunz multiplicity, and, even better, recursive formulae for the whole Hilbert-Kunz series in terms of ideal class groups. There are also striking similarities to the Hasse-Weil Zeta function in arithmetic geometry that are not yet understood.

2. Homological methods.

In addition to the homological questions for rings of positive and mixed characteristic, we intend to include more general homological topics. We will concentrate upon the study of free resolutions, including the tremendous amount of work being done on resolutions of special classes of ideals, and how the structure of the resolution of the coordinate ring of a projective variety reflects the geometry or arithmetic of the variety.

There are in particular two areas of current research activity that we intend to address.

Commutative Algebra and Exterior Algebra.

The first of these areas concerns the interplay between classical commutative algebra and exterior algebra. Although the underlying Bernstein-Gelfand-Gelfand correspondence, which establishes a correspondence between the derived categories of modules over a polynomial ring and a dual exterior algebra, has been known for 25 years, it is only now that the dictionary is sufficiently well understood to bear fruit on classical problems such as resolutions of ideals of points in projective space, syzygies of Veronese or Segre embeddings, Chow forms and resultants. Among the major players in this area are Avramov, Eisenbud, Herzog, S.Popescu, and Schreyer.

An exciting very recent aspect is the appearance of the Koszul complex in the study of D-branes in String Theory, and it is reasonable to expect that in the time leading up to this workshop the connections to classical theory will become clearer and provide for in depth understanding of rather surprising insights into Algebraic Geometry originating from Physics.

Vanishing results for Ext and Tor functors.

A consistent thread in the development of the homological theory of commutative rings has been conditions for Ext and Tor functions to vanish. Although these topics have been studied extensively, it came as a major surprise recently that vanishing of Ext over complete intersections is symmetric in its arguments, and there are strong indications that similar results may hold over general Gorenstein rings. This active area of research, pursued by Avramov, Buchweitz, Huneke, Claudia Miller and their collaborators, is remarkable in that it exposes basic questions about the structure of ubiquitous homological functors. Some of these fundamental problems were raised a long time ago by Auslander and Reiten in connection with the so-called Generalized Nakayama Conjecture, which states that the vanishing of certain Ext modules forces a module to be projective. Recent work in representation theory suggests that deeper understanding will require the study of Hochschild cohomology and concepts related to the fine structure of derived categories and their equivalences.

3. Integral dependence and integral closures.

The concepts of integral extensions and integral closures of rings are central to much of Commutative Algebra. In this part of the proposal we discuss a generalization of these concepts to ideals. This defines a closure operation for which the closure is in general larger than the tight closure discussed in the first section. Integral closure of ideals is closely related to singularity theory, the study of Rees algebras, and multiplier ideals.

Multiplier ideals and cores.

Multiplier ideals are integrally closed ideals that have been defined in complex algebraic geometry using log resolutions. The theory of multiplier ideals has surprising applications in local commutative algebra, as shown by Ein, Lazarsfeld and Smith. On the other hand, Hara, Watanabe, and their coauthors have established a connection to test ideals, a characteristic p notion arising in tight closure theory. The proposed workshop will help in bringing together the different points of view in this rich subject. Multiplier ideals are also related to cores of ideals, a topic receiving a great deal of attention lately. A better understanding of cores would lead to improved versions of the celebrated Briancon-Skoda theorem and solve a conjecture of Kawamata on the non-vanishing of sections of line bundles. This surprising connection was recently discovered by Smith and her coauthors, and one can expect further exciting developments in the near future.

Rees algebras and singularities.

There has been an abundance of new results about Rees algebras and their structure over the past ten years. Rees algebras are the rings in which integral dependence of ideals can be studied and they are the algebraic objects that appear in the process of resolution of singularities. Rees algebras have been used in Kawasaki's celebrated proof of the existence of Macaulifications, a weak form of resolution of singularities. Kawasaki's work, which builds on research of Goto and his school, has not been widely disseminated. The proposed workshop will serve as a forum for discussion of his ideas and techniques. Another important contribution in singularity theory is Cutkosky's work on Abhyankar's conjecture about local factorization of birational maps between nonsingular varieties. We plan to bring together many of the leaders in this subject, including Teissier who has made substantial progress on the problem of desingularization by using toric methods.

Multiplicities and computational problems.

The concept of integral dependence of ideals and modules is essential in intersection theory and the theory of multiplicities. It plays an important role in equisingularity theory as well, through the work of Gaffney, Kleiman and Teissier. On the other hand there is still no efficient algorithm for computing integral closures of ideals: this computational problem has been a focus of work by Vasconcelos and his coauthors.

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