Commutative Algebra

Frontmatter

Lazarsfeld–Mukai Bundles and Applications

Abstract

We survey the development of the notion of Lazarsfeld-Mukai bundles together with various applications, from the classification of Mukai manifolds to Brill-Noether theory and syzygies of K3 sections. To see these techniques at work, we present a short proof of a result of M. Reid on the existence of elliptic pencils.

Marian Aprodu

Some Applications of Commutative Algebra to String Theory

Abstract

String theory was first introduced as a model for strong nuclear interactions, then reinterpreted as a model for quantum gravity, and then all fundamental physics.

Paul S. Aspinwall

Measuring Singularities with Frobenius: The Basics

Abstract

The multiplicity is an important first step in measuring singularities, but it is too crude to give a good measurement. This paper describes the first steps toward understanding a much more subtle measure of singularities which arises naturally in three different contexts - analytic, algebro-geometric, and finally, algebraic. Miraculously, all three approaches lead to essentially the same measurement of singularities: the log canonical threshold (in characteristic zero) and the closely related F-pure threshold (in characteristic p).

Angélica Benito, Eleonore Faber, Karen E. Smith

Three Flavors of Extremal Betti Tables

Abstract

We discuss extremal Betti tables of resolutions in three different contexts. We begin over the graded polynomial ring, where extremal Betti tables correspond to pure resolutions. We then contrast this behavior with that of extremal Betti tables over regular local rings and over a bigraded ring.

Christine Berkesch, Daniel Erman, Manoj Kummini

p −1-Linear Maps in Algebra and Geometry

Abstract

At least since Habousch’s proof of Kempf’s vanishing theorem, Frobenius splitting techniques have played a crucial role in geometric representation theory and algebraic geometry over a field of positive characteristic. In this article we survey some recent developments which grew out of the confluence of Frobenius splitting techniques and tight closure theory and which provide a framework for higher dimension geometry in positive characteristic. We focus on local properties, i.e. singularities, test ideals, and local cohomology on the one hand and global geometric applicatioms to vanishing theorems and lifting of sections on the other.

Manuel Blickle, Karl Schwede

Castelnuovo–Mumford Regularity of Annihilators, Ext and Tor Modules

Abstract

Let R be a Noetherian homogeneous (e.g. standard graded) ring with local Artinian base ring (in degree 0), and let M and N be two finitely generated graded R-modules. We give various upper bounds for the Castelnuovo-Mumford regularity of the annihilator of M and the modules Ext(M,N) and Tor(M,N) in terms of basic invariants of R, M and N.

Markus Brodmann, Cao Huy Linh, Maria-Helena Seiler

Selections from the Letter-Place Panoply

Abstract

There is a fairly extensive literature on letter-place algebras, but mostly for the edification of those working in algebraic combinatorics. Letter-place algebras have not had much play yet in commutative and homological algebra, so the author thought he would talk about them here and perhaps arouse a bit of interest in that subject.

David A. Buchsbaum

Koszul Algebras and Regularity

Abstract

This is a survey paper on commutative Koszul algebras and Castelnuovo-Mumford regularity. Koszul algebras, originally introduced by Priddy, are graded K-algebras R whose residue field K has a linear free resolution as an R-module. The Castelnuovo-Mumford regularity is, after Krull dimension and multiplicity, perhaps the most important invariant of a finitely generated graded module M, as it controls the vanishing of both syzygies and the local cohomology modules of M.

Aldo Conca, Emanuela De Negri, Maria Evelina Rossi

Powers of Ideals: Betti Numbers, Cohomology and Regularity

Abstract

The aim of this paper is to provide an approach to some advances over the last decade concerning homological invariants of powers of a graded ideal that derive from finiteness properties of the Rees algebra of the ideal.

Marc Chardin

Some Homological Properties of Modules over a Complete Intersection, with Applications

Abstract

We survey some recent developments in understanding homological properties of finitely generated modules over a complete intersection. These properties mainly concern with vanishing patterns of Ext and Tor functors. We focus on applications to related areas and open questions.

Hailong Dao

Powers of Square-Free Monomial Ideals and Combinatorics

Abstract

We survey research relating algebraic properties of powers of squarefree monomial ideals to combinatorial structures. In particular, we describe how to detect important properties of (hyper)graphs by solving ideal membership problems and computing associated primes. This work leads to algebraic characterizations of perfect graphs independent of the Strong Perfect Graph Theorem. In addition, we discuss the equivalence between the Conforti-Cornuéjols conjecture from linear programming and the question of when symbolic and ordinary powers of squarefree monomial ideals coincide.

Christopher A. Francisco, Huy Tài Hà, Jeffrey Mermin

A Brief History of Order Ideals

Abstract

The notion of order ideal is no doubt implicit in a paper by Serre in 1958 on free summands of projective modules. A formal definition is given by Bass. However, any algebraist contemplating the question, “on what locus of prime ideals in Spec(R) does an element e in a module E generate a free summand?”, has in fact encountered the concept of an order ideal. In the account on order ideals and their applications in this paper, it is our intent to elaborate on four basic theorems - as we see them - that give insight into the height properties of these ideals. We do this both from a historical view as well as a view of their utility.

E. Graham Evans, Phillip Griffith

Moduli of Abelian Varieties, Vinberg θ-Groups, and Free Resolutions

Abstract

We present a systematic approach to studying the geometric aspects of Vinberg θ-representations. The main idea is to use the Borel-Weil construction for representations of reductive groups as sections of homogeneous bundles on homogeneous spaces, and then to study degeneracy loci of these vector bundles. Our main technical tool is to use free resolutions as an “enhanced” version of degeneracy loci formulas. We illustrate our approach on several examples and show how they are connected to moduli spaces of Abelian varieties. To make the article accessible to both algebraists and geometers, we also include background material on free resolutions and representation theory.

Laurent Gruson, Steven V. Sam, Jerzy Weyman

F-Purity, Frobenius Splitting, and Tight Closure

Abstract

Several applications of the technique of studying when the Frobenius endomorphism from a ring of positive prime characteristic to itself splits are discussed. These include some problems that, historically, motivated the development of the theory. One of these is the theorem that rings of invariants of linearly reductive groups acting on regular rings are Cohen-Macaulay, including normal rings generated by monomials. Another is the characterization of when Stanley-Reisner rings are Cohen-Macaulay. Another is the proof of the existence of finitely generated maximal Cohen-Macaulay modules for graded rings of positive prime characteristic when the normalization has an isolated singularity (or an isolated non-Cohen-Macaulay point), which includes all three-dimensional graded domains of positive characteristic. Applications of the closely related theory of tight closure are also discussed.

Melvin Hochster

Hilbert–Kunz Multiplicity and the F-Signature

Abstract

This paper is a much expanded version of two talks given in Ann Arbor in May of 2012 during the computational workshop on F-singularities. We survey some of the theory and results concerning the Hilbert-Kunz multiplicity and F-signature of positive characteristic local rings.

Craig Huneke

Pure O-Sequences: Known Results, Applications, and Open Problems

Abstract

This paper presents a discussion of the algebraic and combinatorial aspects of the theory of pure O-sequences. Various instances where pure O-sequences appear are described. Several open problems that deserve further investigation are also presented.

Juan Migliore, Uwe Nagel, Fabrizio Zanello

Bounding Projective Dimension

Abstract

This paper is a survey of progress on Stillman’s Question: Let J be a homogeneous ideal in a standard graded polynomial ring over a field. Is there a bound on the projective dimension of J depending only on the number of elements in a minimal system of homogenoeus generators of J and their degrees (in particular, independent of the number of variables)?

Jason McCullough, Alexandra Seceleanu

Brauer–Thrall Theory for Maximal Cohen–Macaulay Modules

Abstract

The Brauer-Thrall Conjectures, now theorems, were originally stated for finitely generated modules over a finite-dimensional k-algebra. They say, roughly speaking, that infinite representation type implies the existence of lots of indecomposable modules of arbitrarily large k-dimension. These conjectures have natural interpretations in the context of maximal Cohen-Macaulay modules over Cohen-Macaulay local rings. This is a survey of progress on these transplanted conjectures.

Graham J. Leuschke, Roger Wiegand

Tight Closure’s Failure to Localize - a Self-Contained Exposition

Abstract

Brenner and Monsky have given a negative solution to the localization problem for tight closure. Here I give a treatment of our counterexample that uses only linear algebra, material from an introductory abstract algebra course, and a little local cohomology developed ab initio. But most of this machinery, useful as it is for understanding the counterexample, may be dispensed with; in this paper the author gives a treatment of the example, using only linear algebra, material from an introductory abstract algebra course, and a little local cohomology developed ab initio.

Paul Monsky

Introduction to the Hyperdeterminant and to the Rank of Multidimensional Matrices

Abstract

The classical theory of determinants was placed on a solid basis by Cayley in 1843. A few years later, Cayley himself elaborated a generalization to the multidimensional setting in two different ways. There are indeed several ways to generalize the notion of determinant to multidimensional matrices. Cayley’s second attempt has a geometric flavor and was very fruitful. This invariant constructed by Cayley is named today hyperdeterminant and reduces to the determinant in the case of square matrices. The explicit computation of the hyperdeterminant presented from the very beginning exceptional difficulties. In 1992, thanks to a fundamental paper by Gelfand, Kapranov and Zelevinsky, the theory was placed in the modern language and many new results have been found. In this survey we introduce the hyperdeterminants and some of its properties from scratch. Our aim is to provide elementary arguments, when they are available. The main tools we use are the biduality theorem and the language of vector bundles.

Giorgio Ottaviani

Commutative Algebra of Subspace and Hyperplane Arrangements

Abstract

Arrangements of linear subspaces have connections with a wealth of mathematical objects in areas as diverse as topology, invariant theory, combinatorics, algebraic geometry, and statistics. Arrangements have also recently played a prominent role in applied mathematics, appearing as key players in data mining and generalized principal component analysis, in the study of the topological complexity of robot motion planning, and in the study of configuration spaces and the Gaudin model of mathematical physics. We give an overview of a number of problems having close connections to commutative algebra and algebraic geometry; the field is very broad so this survey is selective.

Hal Schenck, Jessica Sidman

Cohomological Degrees and Applications

Abstract

This paper is an overview of several cohomological extensions of the ordinary multiplicity function of local algebra. It emphasizes the construction of such functions and the development of their main properties. A select set of applications is used to illustrate their usefulness.

Wolmer V. Vasconcelos