Craig L. Huneke

University of Virginia | UVa · Department of Mathematics

We study the question of which rings, and which families of ideals, have uniform symbolic topologies. In particular, we show that the uniform symbolic topology property holds for all dimension one primes in any normal complete local domain, provided dimension one primes in hypersurfaces have the uniform symbolic topology property. We also discuss b...

If I I is an ideal in a Gorenstein ring S S , and S / I S/I is Cohen-Macaulay, then the same is true for any linked ideal I ′ I’ ; but such statements hold for residual intersections of higher codimension only under restrictive hypotheses, not satisfied even by ideals as simple as the ideal L n L_{n} of minors of a generic 2 × n 2 \times n matrix w...

We provide a natural criterion which implies equality of the finitistic test ideal and test ideal in local rings of prime characteristic. Most notably, we show that the criterion is met by every local weakly $F$-regular ring whose anti-canonical algebra is Noetherian on the punctured spectrum.

Let $(R,\mathfrak{m},\mathbb{k})$ be an equicharacteristic one-dimensional complete local domain over an algebraically closed field $\mathbb{k}$ of characteristic 0. R. Berger conjectured that R is regular if and only if the universally finite module of differentials $\Omega_R$ is a torsion-free $R$-module. We give new cases of this conjecture by e...

Let (R,m,k) be an equicharacteristic one-dimensional complete local domain over an algebraically closed field k of characteristic 0. R. Berger conjectured that R is regular if and only if the universally finite module of differentials ΩR is a torsion-free R module. We give new cases of this conjecture by extending works of Güttes (Arch Math 54:499–...

We explore the classical Lech's inequality relating the Hilbert–Samuel multiplicity and colength of an m-primary ideal in a Noetherian local ring (R,m). We prove optimal versions of Lech's inequality for sufficiently deep ideals in characteristic p>0, and we conjecture that they hold in all characteristics. Our main technical result shows that if (...

Building on previous work by the same authors, we show that certain ideals defining Gorenstein rings have expected resurgence, and thus satisfy the stable Harbourne Conjecture. In prime characteristic, we can take any radical ideal defining a Gorenstein ring in a regular ring, provided its symbolic powers are given by saturations with the maximal i...

If I is an ideal in a Gorenstein ring S and S/I is Cohen-Macaulay, then the same is true for any linked ideal I'. However, such statements hold for residual intersections of higher codimension only under very restrictive hypotheses, not satisfied even by ideals as simple as the ideal L_n of minors of a generic 2 x n matrix when n>3. In this paper w...

We explore the classical Lech's inequality relating the Hilbert--Samuel multiplicity and colength of an $\mathfrak{m}$-primary ideal in a Noetherian local ring $(R,\mathfrak{m})$. We prove optimal versions of Lech's inequality for sufficiently deep ideals in characteristic $p>0$, and we conjecture that they hold in all characteristics. Our main tec...

We give explicit criteria that imply the resurgence of a self-radical ideal in a regular ring is strictly smaller than its codimension, which in turn implies that the stable version of Harbourne's conjecture holds for such ideals. This criterion is used to give several explicit families of such ideals, including the defining ideals of space monomia...

In this paper we prove that, under mild conditions, an equicharacteristic integrally closed domain which is a finite abelian extension of a regular domain has the uniform symbolic topology property.

We investigate the existence of ideals $I$ in a one-dimensional Gorenstein local ring $R$ satisfying $\mathrm{Ext}^{1}_{R}(I,I)=0$.

We investigate the existence of ideals $I$ in a one-dimensional Gorenstein local ring $R$ satisfying $\mathrm{Ext}^{1}_{R}(I,I)=0$.

We survey classical and recent results on symbolic powers of ideals. We focus on properties and problems of symbolic powers over regular rings, on the comparison of symbolic and regular powers, and on the combinatorics of the symbolic powers of monomial ideals. In addition, we present some new results on these aspects of the subject.

We study the structure of $D$-modules over a ring $R$ which is a direct summand of a polynomial or a power series ring $S$ with coefficients over a field. We relate properties of $D$-modules over $R$ to $D$-modules over $S$. We show that the localization $R_f$ and the local cohomology module $H^i_I(R)$ have finite length as $D$-modules over $R$. Fu...

We study conjectured generalizations of a formula of Lech which relates the multiplicity of a finite colength ideal in an equicharacteristic local ring to its colength, and prove one of these generalizations involving the multiplicity of the maximal ideal times the finite colength ideal. We also propose a Lech-type formula that relates multiplicity...

We study conjectured generalizations of a formula of Lech which relates the multiplicity of a finite colength ideal in an equicharacteristic local ring to its colength, and prove one of these generalizations involving the multiplicity of the maximal ideal times the finite colength ideal. We also propose a Lech-type formula that relates multiplicity...

Motivated by Stillman's question, we show that the projective dimension of an ideal generated by four quadric forms in a polynomial ring is at most 6; moreover, this bound is tight. We achieve this bound, in part, by giving a characterization of the low degree generators of ideals primary to height three primes of multiplicities one and two.

We survey classical and recent results on symbolic powers of ideals. We focus on properties and problems of symbolic powers over regular rings, on the comparison of symbolic and regular powers, and on the combinatorics of the symbolic powers of monomial ideals. In addition, we present some new results on these aspects of the subject.

Given a radical ideal $I$ in a regular ring $R$, the Containment Problem of symbolic and ordinary powers of $I$ consists of determining when the containment $I^{(a)} \subseteq I^b$ holds. By work of Ein–Lazersfeld–Smith, Hochster–Huneke and Ma–Schwede, there is a uniform answer to this question, but the resulting containments are not necessarily be...

Given a radical ideal $I$ in a regular ring $R$, the Containment Problem of symbolic and ordinary powers of $I$ consists of determining when the containment $I^{(a)} \subseteq I^b$ holds. By work of Ein-Lazersfeld-Smith, Hochster-Huneke and Ma-Schwede, there is a uniform answer to this question, but the resulting containments are not necessarily be...

We study the structure of $D$-modules over a ring $R$ which is a direct summand of a polynomial or a power series ring $S$ with coefficients over a field. We relate properties of $D$-modules over $R$ to $D$-modules over $S$. We show that the localization $R_f$ and the local cohomology module $H^i_I(R)$ have finite length as $D$-modules over $R$. Fu...

This paper answers in the affirmative a question raised by Karl Schwede concerning an upper bound on the multiplicity of F-pure rings.

We study the behavior of rings with uniform symbolic topologies with respect to finite extensions.

Let $(R,\mathfrak{m},K)$ be a local ring, and let $M$ be an $R$-module of finite length. We study asymptotic invariants, $\beta^F_i(M,R),$ defined by twisting with Frobenius the free resolution of $M$. This family of invariants includes the Hilbert-Kunz multiplicity ($e_{HK}(\mathfrak{m},R)=\beta^F_0(K,R)$). We discuss several properties of these n...

Let $R$ be a formal power series ring over a field, with maximal ideal $\mathfrak m$, and let $I$ be an ideal of $R$ such that $R/I$ is Artinian. We study the iterated socles of $I$, that is the ideals which are defined as the largest ideal $J$ with $J\mathfrak m^s\subset I$ for a fixed positive integer $s$. We are interested in these ideals in con...

We investigate symmetry in the vanishing of Ext for finitely generated modules over local Gorenstein rings. In particular, we define a class of local Gorenstein rings, which we call AB rings, and show that for finitely generated modules $M$ and $N$ over an AB ring $R$, $Ext^i_R(M,N)=0$ for all $i >> 0$ if and only if $Ext^i_R(N,M)=0$ for all $i >>...

This paper summarizes three talks given by the author during a PASI conference in Olinda, Brasil

This is a survey article on Hilbert-Kunz multiplicity and the F-signature.

These notes are based on three lectures given by the first author as part of an introductory workshop at MSRI for the program in Commutative Algebra, 2012-13. The notes follow the talks, but there are extra comments and explanations, as well as a new section on the uniform Artin-Rees theorem. The notes deal with the theme of uniform bounds, both ab...

We prove that for all $n$, simultaneously, we can choose prime filtrations of $R/I^n$ such that the set of primes appearing in these filtrations is finite.

Motivated by Stillman's question, we show that the projective dimension of an ideal generated by four quadric forms in a polynomial ring has projective dimension at most 9.

Let $R$ be a polynomial ring over a field. We prove an upper bound for the multiplicity of $R/I$ when $I$ is a homogeneous ideal of the form $I=J+(F)$, where $J$ is a Cohen-Macaulay ideal and $F\notin J$. The bound is given in terms of two invariants of $R/J$ and the degree of $F$. We show that ideals achieving this upper bound have high depth, and...

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. © 2013 Springer Science+Business Media New York. All rights reserved.

This paper answers in the affirmative a question raised by Karl Schwede concerning an upper bound on the multiplicity of F-pure rings.

We prove a sharp upper bound for the projective dimension of ideals of height two generated by quadrics in a polynomial ring with arbitrary large number of variables.

Motivated by a question of Stillman, we find a sharp upper bound for the projective dimension of ideals of height two generated by quadrics. In a polynomial ring with arbitrary large number of variables, we prove that ideals generated by n quadrics define cyclic modules with projective dimension at most 2n - 2. We refine this bound according to the...

Let $K$ be an algebraically closed field. There has been much interest in characterizing multiple structures in $\P^n_K$ defined on a linear subspace of small codimension under additional assumptions (e.g. Cohen-Macaulay). We show that no such finite characterization of multiple structures is possible if one only assumes Serre's $(S_1)$ property ho...

Let $S$ be a positively graded polynomial ring over a field of characteristic 0, and $I\subset S$ a proper graded ideal. In this note it is shown that $S/I$ is Golod if $\partial(I)^2\subset I$. Here $\partial(I)$ denotes the ideal generated by all the partial derivatives of elements of $I$. We apply this result to find large classes of Golod ideal...

In this paper we give new lower bounds on the Hilbert-Kunz multiplicity of unmixed nonregular local rings, bounding them uniformly away from 1. Our results improve previous work of Aberbach and Enescu.

We study a notion called n-standardness (defined by M.E. Rossi (2000) in [10] and extended in this paper) of ideals primary to the maximal ideal in a Cohen–Macaulay local ring and some of its consequences. We further study conditions under which the maximal ideal is 3-standard, first proving results for when the residue field has prime characterist...

In this paper we give new upper bounds on the regularity of edge ideals whose resolutions are k-steps linear; surprisingly, the bounds are logarithmic in the number of variables. We also give various bounds for the projective dimension of such ideals, generalizing other recent results. By Alexander duality, our results also apply to unmixed square-...

This paper gives new bounds on the first Hilbert coefficient of an ideal of finite colength in a Cohen-Macaulay local ring. The bound given is quadratic in the multiplicity of the ideal. We compare our bound to previously known bounds, and give examples to show that at least in some cases it is sharp. The techniques come largely from work of Elias,...

Searching for structural reasons behind old results and conjectures of Chudnovksy regarding the least degree of a nonzero form in an ideal of fat points in projective N-space, we make conjectures which explain them, and we prove the conjectures in certain cases, including the case of general points in the projective plane. Our conjectures were also...

In this paper we give new lower bounds on the Hilbert-Kunz multiplicity of unmixed non-regular local rings, bounding them uniformly away from one. Our results improve previous work of Aberbach and Enescu.

Let R be a Cohen-Macaulay ring and M a maximal Cohen-Macaulay R-module. Inspired by recent striking work by Iyama, Burban-Iyama-Keller-Reiten and Van den Bergh we study the question of when the endomorphism ring of M has finite global dimension via certain conditions about vanishing of $\Ext$ modules. We are able to strengthen certain results by Iy...

We study a notion called $n$-standardness (defined by M. E. Rossi and extended in this paper) of ideals primary to the maximal ideal in a Cohen-Macaulay local ring and some of its consequences. We further study conditions under which the maximal ideal is three-standard, first proving results when the residue field has prime characteristic and then...

These notes are based on five lectures given at the Summer School on Commutative Algebra held at the CRM in Barcelona during July, 1996. I would like to thank the organizers J. Elias, J. M. Giral, R. M. Miró-Roig, and S. Zarzuela for the excellent job they did. The great success of the Summer School was due mainly to their efforts.

In this paper we discuss various refinements and generalizations of a theorem of Sankar Dutta and Paul Roberts. Their theorem gives a criterion for $d$ elements in a $d$-dimensional Noetherian Cohen-Macaulay local ring to be a system of parameters, i.e., to have height $d$. We chiefly remove the assumption that the ring be Cohen-Macaulay and discus...

The $F$-threshold $c^J(\a)$ of an ideal $\a$ with respect to an ideal $J$ is a positive characteristic invariant obtained by comparing the powers of $\a$ with the Frobenius powers of $J$. We study a conjecture formulated in an earlier paper \cite{HMTW} by the same authors together with M. Musta\c{t}\u{a}, which bounds $c^J(\a)$ in terms of the mult...

The quasilength of a finitely generated module that is killed by a power of a finitely generated ideal I is introduced: it is the length of a shortest filtration of the module with factors that are cyclic modules killed by I. This notion is then used to define a notion of content for the dth local cohomology module of a ring or module with support...

We examine the question of whether the support of an arbitrary local coho- mology module of a finitely generated module over a Noetherian ring with support in a given ideal must be closed in the Zariski topology. Several results giving an armative answer to this question are given; in particular, we show that the support is closed when the given id...

Let (R, m) be a local ring. We study the question of when there exists a positive integer h such that for all prime ideals P⊆R, the symbolic power P(hn) is contained in Pn, for all n≥1. We show that such an h exists when R is a reduced isolated singularity such that R either contains a field of positive characteristic and R is F-finite or R is esse...

We study the linkage classes of homogeneous ideals in polynomial rings. An ideal is said to be homogeneously licci if it can be linked to a complete intersection using only homogeneous regular sequences at each step. We ask a natural question: if $I$ is homogeneously licci, then can it be linked to a complete intersection by linking using regular s...

The F-threshold $c^J(\a)$ of an ideal $\a$ with respect to the ideal $J$ is a positive characteristic invariant obtained by comparing the powers of $\a$ with the Frobenius powers of $J$. We show that under mild assumptions, we can detect the containment in the integral closure or the tight closure of a parameter ideal using F-thresholds. We formula...

This note proves that if S is an unramified regular local ring and I, J proper ideals of height at least two, then S/IJ is never Gorenstein.

Let R be a local Noetherian domain of positive characteristic. A theorem of Hochster and Huneke [M. Hochster, C. Huneke, Infinite integral extensions and big Cohen–Macaulay algebras, Ann. of Math. 135 (1992) 53–89] states that if R is excellent, then the absolute integral closure of R is a big Cohen–Macaulay algebra. We prove that if R is the homom...

We give a simple algorithm to decide whether a monomial ideal of finite colength in a polynomial ring is licci , that is, in the linkage class of a complete intersection. The algorithm proves that whether or not such an ideal is licci does not depend on whether we restrict the linkage by allowing only monomial regular sequences, or homogeneous regu...

A fundamental property connecting the symbolic powers and the usual powers of ideals in regular rings was discovered by Ein, Lazarsfeld, and Smith in 2001, and later extended by Hochster and Huneke in 2002. In this paper we give further generalizations which give better results in case the quotient of the regular ring by the ideal is F-pure or F-pu...

This note makes a correction to the paper “Tensor products of modules and the ridigity of Tor” [Math. Ann. 299, No. 3, 449–476 (1994; Zbl 0803.13008)], a correction which is needed due to an incorrect convention for the depth of the zero module.

This article is based on five lectures the author gave during the summer school, Interactions between homotopy theory and algebra, from July 26-August 6, 2004, held at the University of Chicago, organized by Lucho Avramov, Dan Christensen, Bill Dwyer, Mike Mandell, and Brooke Shipley. These notes introduce basic concepts concerning local cohomology...

Without Abstract

This paper studies Frobenius powers of parameter ideals in a commutative Noetherian local ring R of prime characteristic p. For a given ideal a of R, there is a power Q of p, depending on a, such that the Qth Frobenius power of the Frobenius closure of a is equal to the Qth Frobenius power of a. The paper addresses the question as to whether there...

This paper studies Frobenius powers of parameter ideals in a commutative Noetherian local ring $R$ of prime characteristic $p$. For a given ideal $\fa$ of $R$, there is a power $Q$ of $p$, depending on $\fa$, such that the $Q$-th Frobenius power of the Frobenius closure of $\fa$ is equal to the $Q$-th Frobenius power of $\fa$. The paper addresses t...

We introduce classes of rings which are close to being Gorenstein. These rings arise naturally as specializations of rings of countable CM type. We study these rings in detail, and along the way generalize an old result of Teter which characterized Artinian rings which are Gorenstein rings modulo their socle.

Let R be a local Noetherian domain of positive characteristic. A theorem of Hochster and Huneke (1992) states that if R is excellent, then the absolute integral closure of R is a big Cohen-Macaulay algebra. We prove that if R is the homomorphic image of a Gorenstein local ring, then all the local cohomology (below the dimension) of such a ring maps...

In this paper we prove a finiteness result for infinite minimal free resolutions over a Noetherian local ring R: If M is a module, such as the residue field, that is locally free of constant rank on the punctured spectrum of R, and I⊂R is an ideal, then the maps fn:Fn→Fn-1 in the minimal free resolution of M satisfy the uniform Artin–Rees property:...

This paper gives criteria for a Cohen–Macaulay local ring to be Gorenstein, in terms of the vanishing of Ext modules. The main results extend earlier work of Ulrich.

The workshop on Commutative Algebra was very well attended by the important senior researchers in the field and many promising young mathematicians. A major subgroup of the participants was formed by researchers in affine geometry, a neighboring field that has strong interactions with commutative algebra. The NSF Oberwolfach program made it possibl...

Let S=K[x_1,..., x_n], let A,B be finitely generated graded S-modules, and let m=(x_1,...,x_n). We give bounds for the Castelnuovo-Mumford regularity of the local cohomology of Tor_i(A,B) under the assumption that the Krull dimension of Tor_1(A,B) is at most 1. We apply the results to syzygies, Groebner bases, products and powers of ideals, and to...

Our focus in this paper is in effective computation of the core core(I) of an ideal I which is defined to be the intersection of all minimal reductions of I. The first main result is a closed formula for the graded core(m) of the maximal graded ideal m of an arbitrary standard graded algebra A over a field k. This formula allows us to study basic p...

This paper contains two theorems concerning the theory of maximal Cohen--Macaulay modules. The first theorem proves that certain Ext groups between maximal Cohen--Macaulay modules $M$ and $N$ must have finite length, provided only finitely many isomorphism classes of maximal Cohen--Macaulay modules exist having ranks up to the sum of the ranks of $...

Let (R,m,k) be an excellent, local, normal ring of characteristic p with a perfect residue field and dim R=d. Let M be a finitely generated R-module. We show that there exists a real number beta(M) such that lambda(M/I^[q]M) = e_{HK}(M) q^d + beta(M) q^{d-1} + O(q^{d-2}).

In studying Nakayama’s 1958 conjecture on rings of infinite dominant dimension, Auslander and Reiten proposed the following generalization: Let Λ be an Artin algebra and M a Λ-generator such that Ext i Λ(M, M) = 0 for all i≥1; then M is projective. This conjecture makes sense for any ring. We establish Auslander and Reiten’s conjecture for excellen...

We introduce classes of rings which are close to being Gorenstein. These rings arise naturally as specializations of rings of countable CM type. We study these rings in detail, and along the way generalize an old result of Teter which characterized Artinian rings which are Gorenstein rings modulo their socle.

We discuss vanishing of cohomology of finite modules over Cohen-Macaulay local rings $(R, \mathfrak m)$. Special attention is given to the case when the modules are annihilated by $\mathfrak m^2$. (Note that if $\mathfrak m^3=0$, then we can assume the modules satisfy this condition.) In this case we obtain effective versions of conjectures of Ausl...

This article outgrew from an effort to understand our basic question: Are the annihilators of the non-zero Koszul homology modules $H_i$ of an unmixed ideal $I$ contained in the integral closure $\bar{I}$ of $I$? We also obtain some variations on a result of Burch, which continue the theme of the paper in that they deal with annihilators of homolog...

We study the notion of special tight closure of an ideal and show that it can be used as a tool for tight closure computations.

In studying Nakayama's 1958 conjecture on rings of infinite dominant dimension, Auslander and Reiten proposed the following generalization: Let Lambda be an Artin algebra and M a Lambda-generator such that Ext^i_Lambda(M,M)=0 for all i \geq 1; then M is projective. This conjecture makes sense for any ring. We establish Auslander and Reiten's conjec...

We show in this paper that the Briancon-Skoda theorem holds for all ideals in F-rational rings of positive prime characteristic, and also in rings with rational singularities which are of finite type over a field of characteristic 0. Moreover, in Gorenstein F-rational rings of characteristic p we show that in many cases the bound given in the Brian...

The purpose of this paper is to prove a generalization of Faltings' connectedness theorem which asserts that, for a complete local domain R of dimension n, the punctured spectrum of R/I is connected if the ideal I is generated by at most n-2 elements. We replace the condition that R be a domain by the requirement that the canonical module of R be i...

In this paper we study various equivalent conditions for tight closure to commute with localization. If N is a submodule of a finitely generated module M over a Noetherian commutative ring of characteristic p, then a test exponent for c,N,M is defined to be a power q' of p such that u is in the tight closure of N in M whenever cu^q is in the qth Fr...

In this paper we generalize the theorem of Ein-Lazarsfeld-Smith (concerning the behavior of symbolic powers of prime ideals in regular rings finitely generated over a field of characteristic 0) to arbitrary regular rings containing a field. The basic theorem states that in such rings, if P is a prime ideal of height c, then for all n, the symbolic...

We study content ideals of polynomials and their behavior under multiplication. We give a generalization of the Lemma of Dedekind-Mertens and prove the converse under suitable dimensionality restrictions.

We give a new and simple proof that unmixed local rings having Hilbert-Kunz multiplicity equal to 1 must be regular.

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, am...

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 Kwieci\'nski, which estimates the height of certain Fitting ideals of modules having an equid...

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...

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 sh...