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Let I be an ideal in a polynomial ring over a perfect field. We give new methods for comput ing the equidimensional parts and radical of I, for localizing I with respect to another ideal, and thus for finding the pr imary decomposit ion of / . Our methods rest on modern ideas from commutat ive algebra, and are direct in the sense that they avoid the generic… (More)

- Melvin Hochster, Craig Huneke
- 2002

All given rings in this paper are commutative, associative with identity, and Noetherian. Recently, L. Ein, R. Lazarsfeld, and K. Smith [ELS] discovered a remarkable and surprising fact about the behavior of symbolic powers of ideals in affine regular rings of equal characteristic 0: if h is the largest height of an associated prime of I , then I (hn) ⊆ I n… (More)

- Luchezar L. Avramov, Joan E. Elias, +7 authors Luchezar L. Avramov
- 2009

This text is based on the notes for a series of five lectures to the Barcelona Summer School in Commutative Algebra at the Centre de Recerca Matemàtica, Institut d’Estudis Catalans, July 15–26, 1996. Joan Elias, José Giral, Rosa Maria Miró-Roig, and Santiago Zarzuela were successively fantastic organizers, graceful hosts, and tactful editors. I am extremely… (More)

- Craig Huneke, Graham J. Leuschke
- 2008

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… (More)

- Craig Huneke, Amelia Taylor, +4 authors Amelia Taylor
- 2006

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, and use… (More)

- Craig Huneke, GRAHAM J. LEUSCHKE
- 2008

We prove (the excellent case of) Schreyer’s conjecture that a local ring with countable CM 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. Let (R,m) be a (commutative Noetherian) local ring of dimension d. Recall that a nonzero… (More)

an isomorphism? These questions have interest partly because if S and .S’(J, S) are Cohen-Macaulay, then so is gr, S := S/J@ J/J’... [ 16 1 and under these hypotheses if N is perfect and R @ .S(J, S) = .$(I, R), then R and .7(1. R) are Cohen-Macaulay too. Thus, gr,R is Cohen-Macaulay, and its torsion freeness and normality, for exmple, can be characterized… (More)

Let S = K[x1, . . . , xn], let A, B be finitely generated graded S-modules, and let m = (x1, . . . , xn) ⊂ S. We give bounds for the regularity of the local cohomology of Tork (A, B) in terms of the graded Betti numbers of A and B, under the assumption that dim Tor1 (A, B) ≤ 1. We apply the results to syzygies, Gröbner bases, products and powers of ideals,… (More)

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… (More)