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Craig Huneke

Publications and source records attributed to Craig Huneke.

64 records · Page 4Linked to original sources
arXiv2002

On the integral closure of ideals

Among the several types of closures of an ideal $I$ that have been defined and studied in the past decades, the integral closure $\bar{I}$ has a central place being one of the earliest and most relevant. Despite this role, it is often a difficult challenge to describe it concretely once the generators of $I$ are known. Our aim in this note is to show that in a broad class of ideals their radicals play a fundamental role in testing for integral closedness, and in case $I\neq \bar{I}$, $\surd{I}$ is still helpful in finding some fresh new elements in $\bar{I}\setminus I$. Among the classes of ideals under consideration are: complete intersection ideals of codimension two, generic complete intersection ideals, and generically Gorenstein ideals.

math.AC↗
arXiv2002

A cancellation theorem for ideals

We prove cancellation theorems for special ideals in Gorenstein local rings. These theorems take the form that if KI is contained in JI, then K is contained in J.

math.AC↗
arXiv2002

Hyman Bass and Ubiquity: Gorenstein Rings

This paper is based on a talk given by the author in October, 1997 at a conference at Columbia University in celebration of Hyman Bass's 65th birthday. The paper details some of the history of Gorenstein rings and their uses.

math.AC↗
arXiv2002

Heights of Ideals of Minors

The classical "generalized principal ideal theorems" of Macaulay, Eagon-Northcott, and others give sharp bounds on the heights of determinantal ideals in arbitrary rings. But in regular local rings (or graded polynomial rings) these are far from sharp, and various questions about vector bundles, as well as other questions in commutative algebra, amount to asking what the real bounds are. We give partial answers, and more generally we prove new height bounds in local rings of given embedding codimension. Our theorems extend and sharpen results of Faltings and Bruns.

math.AC↗
arXiv2002

A Simple Proof of some Generalized Principal Ideal Theorems

Using symmetric algebras we simplify (and slightly strengthen) the Bruns-Eisenbud-Evans "generalized principal ideal theorem" on the height of order ideals of non-minimal generators in a module. We also obtain a simple proof and an extension of a result by Kwieciński, which estimates the height of certain Fitting ideals of modules having an equidimensional symmetric algebra.

math.AC↗
arXiv2002

What is the Rees Algebra of a Module?

In this paper we show that the Rees algebra can be made into a functor on modules over a ring in a way that extends its classical definition for ideals. The Rees algebra of a module M may be computed in terms of a "maximal" map f from M to a free module. It is the image of the map induced by f on symmetric algebras. We show that the analytic spread and reductions of M can be determined from any embedding of M into a free module, and in characteristic 0--but not in positive characteristic!--the Rees algebra itself can be computed from any such embedding.

math.AC↗
arXiv2002

Order Ideals and a Generalized Krull Height Theorem

Let N be a finitely generated module over a Noetherian local ring (R,m). We give criteria for the height of the order ideal N^*(x) of an element x \in N to be bounded by the rank of N. The Generalized Principal Ideal Theorem of Bruns, Eisenbud and Evans says that this inequality always holds if x \in mN, but if x is a minimal generator the question is more subtle; for example, when R is graded and N represents a vector bundle on Proj R, the question is connected with the vanishing of the top chern class. We show that the inequality does hold if the hypothesis x \in mN becomes true after first extending scalars to some local domain and then factoring out torsion. We also give conditions under which it holds in terms of residual intersections and integral closures of modules. We derive information about order ideals that leads to bounds on the heights of trace ideals of modules--even in circumstances where we do not have the expected bounds for the heights of the order ideals.

math.AC↗
arXiv2002

Local rings of countable Cohen--Macaulay type

We prove (the excellent case of) Schreyer's conjecture that a local ring with countable Cohen--Macaulay type has at most a one-dimensional singular locus. Furthermore we prove that the localization of a Cohen-Macaulay local ring of countable CM type is again of countable CM type.

math.AC↗