arXiv subjects

Craig Huneke

Publications and source records attributed to Craig Huneke.

At least 55 records · Page 3Linked to original sources

arXiv2010

Vanishing of Ext, cluster tilting modules and finite global dimension of endomorphism rings

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 Iyama on connections between a higher dimension version of Auslander correspondence and existence of non-commutative crepant resolutions. We also recover and extend to positive characteristics a recent Theorem by Burban-Iyama-Keller-Reiten on cluster-tilting objects in the category of maximal Cohen-Macaulay modules over reduced 1-dimensional hypersurfaces.

math.AC↗

arXiv2010

What is a system of parameters?

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 discuss similar theorems.

math.AC↗

arXiv2010

Multiplicity bounds in graded rings

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. Mustaţă, which bounds $c^J(\a)$ in terms of the multiplicities $e(\a)$ and $e(J)$, when $\a$ and $J$ are zero-dimensional ideals and $J$ is generated by a system of parameters. We prove the conjecture when $\a$ and $J$ are generated by homogeneous systems of parameters in a Noetherian graded $k$-algebra. We also prove a similar inequality involving, instead of the $F$-threshold, the jumping number for the generalized parameter test submodules introduced in \cite{ST}.

math.AC↗

arXiv2007

F-thresholds, tight closure, integral closure, and multiplicity bounds

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 formulate a conjecture bounding $c^J(\a)$ in terms of the multiplicities $e(\a)$ and $e(J)$, when $\a$ and $J$ are zero-dimensional ideals, and $J$ is generated by a system of parameters. We prove the conjecture when $J$ is a monomial ideal in a polynomial ring, and also when $\a$ and $J$ are generated by homogeneous systems of parameters in a Cohen-Macaulay graded $k$-algebra.

math.AC↗

arXiv2007

Minimal Homogenous Liaison and Licci Ideals

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 sequences of forms of smallest possible degree at each step (we call such ideals minimally homogeneously licci)? In this paper we answer this question in the negative. In particular, for every $n\geq 28$ we construct a set of $n$ points in $\mathbb P^3$ which are homogeneously licci, but not minimally homogeneously licci. Moreover, we prove that one cannot distinguish between the classes of homogeneously licci and non-licci ideals based only on their Hilbert functions, nor distinguish between homogeneously licci and minimally homogeneously licci ideals based solely on the graded Betti numbers. Finally, by taking hypersurface sections, we show that the natural question has a negative answer whenever the height of the ideal is at least three.

math.AC↗

arXiv2006

Frobenius test exponents for parameter ideals in generalized Cohen-Macaulay local rings

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 the question as to whether there exists a {\em uniform} $Q_0$ which `works' in this context for all parameter ideals of $R$ simultaneously. In a recent paper, Katzman and Sharp proved that there does exists such a uniform $Q_0$ when $R$ is Cohen--Macaulay. The purpose of this paper is to show that such a uniform $Q_0$ exists when $R$ is a generalized Cohen--Macaulay local ring. A variety of concepts and techniques from commutative algebra are used, including unconditioned strong $d$-sequences, cohomological annihilators, modules of generalized fractions, and the Hartshorne--Speiser--Lyubeznik Theorem employed by Katzman and Sharp in the Cohen--Macaulay case.

math.AC↗

arXiv2006

Absolute integral closure in positive characteristic

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 to zero in a finite extension of the ring. There results an extension of the original result of Hochster and Huneke to the case in which R is a homomorphic image of a Gorenstein local ring, and a considerably simpler proof of this result in the cases where the assumptions overlap, e.g., for complete Noetherian local domains.

math.AC↗

arXiv2004

Hilbert-Kunz Functions for Normal Rings

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}).

math.AC↗

arXiv2004

The Regularity of Tor and Graded Betti Numbers

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 the relationship of the Rees and Symmetric algebras. For example we show that any homogeneous linearly presented m-primary ideal has some power equal to a power of m; and if the first (roughly) (n-1)/2 steps of the resolution of I are linear, then I^2 is a power of m.

math.AC↗

arXiv2004

On the core of ideals

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 properties of the graded core and to construct counter-examples to some open questions on the core of ideals in a local ring. For instance, we can show that in general, core(m \otimes E) \neq core(m)\otimes E, where E is a field extension of k. From this it follows that the equation core(I R') = core(I)R' does not hold for an arbitrary flat local homomorphism R \to R' of Cohen-Macaulay local rings. The second main result proves the formulae core(I)= (J^r:I^r)I = (J^r:I^r)J = J^{r+1}:I^r for any equimultiple ideal I in a Cohen-Macaulay ring R with with characteristic zero residue field, where J is a minimal reduction of I and r is its reduction number. This result has been obtained independently by Polini-Ulrich and Hyry-Smith in the one-dimensional case or when R is a Gorenstein ring. Moreover, we can prove that core(I) = IK, where K is the conductor of R in the blowing-up ring at I.

math.AC↗

arXiv2004

Two theorems about maximal Cohen--Macaulay modules

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 $M$ and $N$. This has several corollaries. In particular it proves that a Cohen--Macaulay local ring of finite Cohen--Macaulay type has an isolated singularity. A well-known theorem of Auslander gives the same conclusion but requires that the ring be Henselian. Other corollaries of our result include statements concerning when a ring is Gorenstein or a complete intersection on the punctured spectrum, and the recent theorem of Leuschke and Wiegand that the completion of an excellent Cohen--Macaulay local ring of finite Cohen--Macaulay type is again of finite Cohen--Macaulay type. The second theorem proves that a complete local Gorenstein domain of positive characteristic $p$ and dimension $d$ is $F$-rational if and only if the number of copies of $R$ splitting out of $R^{1/p^e}$ divided by $p^{de}$ has a positive limit. This result generalizes work of Smith and Van den Bergh. We call this limit the $F$-signature of the ring and give some of its properties.

math.AC↗

arXiv2004

Rings which are almost Gorenstein

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.

math.AC↗

arXiv2004

Integral closure of ideals and annihilators of homology

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 homology and integrally closed ideals.

math.AC↗

arXiv2003

On a conjecture of Auslander and Reiten

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 conjecture for excellent Cohen--Macaulay normal domains containing the rational numbers, and slightly more generally.

math.AC↗

arXiv2002

F-rational rings and the integral closures of ideals

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 Briancon-Skoda theorem may be improved by at least one.

math.AC↗

arXiv2002

Indecomposable canonical modules and connectedness

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 indecomposable. We also study equivalent conditions for the canonical module to be indecomposable; under mild conditions this is equivalent to the S_2-ification of the local ring to be local.

math.AC↗

arXiv2002

Localization and test exponents for tight closure

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 Frobenius power of N for some q \ge q'. We prove that that a test exponent for a locally stable test element c and for N,M as above exists if and only if the tight closure of N in M commutes with localization. Other equivalent conditions are given for tight closure to commute with localization.

math.AC↗

arXiv2002

Comparison of symbolic and ordinary powers of ideals

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 (cn)th power of P is contained in the nth power of P. Results are also given in the non-regular case: one must correct by a power of the Jacobian ideal in rings where the Jacobian ideal is defined.

math.AC↗