A fundamental property connecting the symbolic powers and the usual powers of ideals in regular r... more 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-pure type. Our methods also give insight into a conjecture of Eisenbud and Mazur concerning the existence of evolutions. The methods used come from tight closure and reduction to positive characteristic.
Proceedings of the American Mathematical Society, 1990
Let ( R , m ) \\left ( {R,m} \\right ) be a complete local domain containing the rationals. If I ... more Let ( R , m ) \\left ( {R,m} \\right ) be a complete local domain containing the rationals. If I ⊆ R I \\subseteq R is a one-fibered ideal then there is a constant l l , depending only on R R and I I , such that if f ∈ m f \\in m and f ∉ I n f \\notin {I^n} , then there exists a derivation d d such that d ( f ) ∉ I n + l d\\left ( f \\right ) \\notin {I^{n + l}} .
Proceedings of the American Mathematical Society, 2011
This paper gives new bounds on the first Hilbert coefficient of an ideal of finite colength in a ... more 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, Rossi, Valla, and Vasconcelos.
Springer Proceedings in Mathematics & Statistics, 2018
We survey classical and recent results on symbolic powers of ideals. We focus on properties and p... more 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.
Proceedings of the American Mathematical Society, 2007
This note proves that if S is an unramified regular local ring and I, J proper ideals of height a... more 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.
CBMS Regional Conference Series in Mathematics, 1996
88 Craig Huneke, Tight closure and its applications. 1996 87 John Erik Forneess, Dynamics in seve... more 88 Craig Huneke, Tight closure and its applications. 1996 87 John Erik Forneess, Dynamics in several complex variables, 1996 86 Sorin Popa, Classification of subfactors and their endomorphisms, 1995 85 Michlo Jimbo and Tetsuji Miwa, ...
We prove that for all n, simultaneously, we can choose prime filtrations of R/I n such that the s... more 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.
Abstract: This article is based on five lectures the author gave during the summer school, Intera... more Abstract: This article is based on five lectures the author gave during the summer school, Interactions between Homotopy Theory and Algebra, from July 26August 6, 2004, held at the University of Chicago, organized by Lucho Avramov, Dan Christensen, Bill Dwyer, ...
In this paper we give new lower bounds on the Hilbert-Kunz multiplicity of unmixed nonregular loc... more 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.
Motivated by Stillman's question, we show that the projective dimension of an ideal generated by ... more 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.
Proceedings of the American Mathematical Society, 2015
Let R be a polynomial ring over a field. We prove an upper bound for the multiplicity of R/I when... more 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 / ∈ J. We show that ideals achieving this upper bound have high depth, and provide a purely numerical criterion for the Cohen-Macaulay property. Applications to quasi-Gorenstein rings and almost complete intersections are given.
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Papers by C. Huneke