Craig Huneke

Direct methods for primary decomposition

SummaryLetI be an ideal in a polynomial ring over a perfect field. We given new methods for computing the equidimensional parts and radical ofI, for localizingI with respect to another ideal, and thus for finding the primary decomposition ofI. Our methods rest on modern ideas from commutative algebra, and are direct in the sense that they avoid the generic projections used by Hermann (1926) and all others until now.Some of our methods are practical for certain classes of interesting problems, and have been implemented in the computer algebra system Macaulay of Bayer and Stillman (1982–1992).

Bass numbers of local cohomology modules

Let A be a regular local ring of positive characteristic. This paper is concerned with the local cohomology modules of A itself, but with respect to an arbitrary ideal of A. The results include that all the Bass numbers of all such local cohomology modules are finite, that each such local cohomology module has finite set of associated prime ideals, and that, whenever such a local cohomology module is Artinian, then it must be injective. (This last result had been proved earlier by Hartshorne and Speiser under the additional assumptions that A is complete and contains its residue field which is perfect.) The paper ends with some low-dimensional evidence related to questions about whether the analogous statements for regular local rings of characteristic 0 are true

Comparison of symbolic and ordinary powers of ideals

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 for all n ≥ 0. Here, if W is the complement of the union of the associated primes of I , I (t) denotes the contraction of I t RW to R, where RW is the localization of R at the multiplicative system W . Their proof depends on the theory of multiplier ideals, including an asymptotic version, and, in particular, requires resolution of singularities as well as vanishing theorems. We want to acknowledge that without their generosity and quickness in sharing their research this manuscript would not exist. Our objective here is to give stronger results that can be proved by methods that are, in some ways, more elementary. Our results are valid in both equal characteristic 0 and in positive prime characteristic p, but depend on reduction to characteristic p. We use tight closure methods and, in consequence, we need neither resolution of singularities nor vanishing theorems that may fail in positive characteristic. For the most basic form of the result, all that we need from tight closure theory is the definition of tight closure and the fact that in a regular ring, every ideal is tightly closed. We note that the main argument here is closely related to a proof given in [Hu, 5.14–16, p. 45] that regular local rings in characteristic p are UFDs,

The theory of d-sequences and powers of ideals

In this paper we study the question of finding depth R/I” given an ideal I in a commutative ring. This problem is difficult in general; even for simple examples depth R/Z* can be difficult to compute. Brodmann has shown that for any commutative Noetherian ring depth R/Z” become stable for large n; finding this value, however, is quite hard. This problem originally arose through consideration of the following example: suppose k is a field and X= (xij) is an n

On the symmetric and Rees algebra of an ideal generated by a d-sequence

Let R be a commutative ring and I an ideal of R. In this paper, we consider the question of when the symmetric algebra of I is a domain, and hence isomorphic to the Rees algebra of I. (see Section 2 for definitions.) Several authors have studied this question (for example, [I, 4, 9, lo], or [14]). In the cases in which the symmetric algebra is a domain, other questions have been asked: Is it Cohen-Macaulay [l] ? Is it factorial [15] ? Is it integrally closed [1, 121 ? In this paper we prove the symmetric algebra of I is a domain whenever R is a domain and I is generated by a d-sequence (see [6] or [7]). A sequence of elements xi ,..., x, in R is said to be a d-sequence if (i) xi 4 (x1 ,..., xiwl , xi+r ,..., x,) for i between 1 and n and (ii) if {il ,..., ii} is a subset (possibly (6) of u,..., TZ} and K, rn~ { l,..., n}\(il ,..., ii} then((xil ,..., xi,) : xlcx,) = ((xi, ,..., xi,) : xlc). Many examples were given in [7] of d-sequences. We list some examples here.

Birational extensions in dimension two and integrally closed ideals

The aim of this paper is to examine, primarily from an algebraic point of view, the structure of a 2-dimensional normal local domain (S, n) which birationally dominates a 2-dimensinal regular local ring (R, m). From the geometric point of view, the sine qua non is Lipman’s paper [Lr] on rational singularities, for it follows from one of the early results in that paper that S has a rational singularity. Using elementary algebraic techniques we are able to recover much information concerning the structure of S. We have particularly focused on the fact (from Lipman [Ll] and Artin [Ar]) that S must have minimal multiplicity. We hope this approach will aid in understanding the algebra of rational singularities and in the exploration of many open questions in dimension two and in higher dimensions. The framework of the paper is set up as follows. In the first section we use basic algebraic techniques such as analytic independence and Zariski’s Main Theorem to prove, for example, that S has a regular height 1 prime if R/m is infinite. If S is not regular, we may assume that R is “maximally regular” in S and when this is the case we show, for instance, that the

LINKAGE AND THE KOSZUL HOMOLOGY OF IDEALS

0. Introduction. Let R be a local ring, and I an ideal of R. Associated to I are several graded algebras: the symmetric algebra of the module I, Sym(I), the Rees algebra of I, defined to be the algebra R (I) = (Gn=O InF, and the associated graded algebra of I, gr1(R) = RII G II/2 (? ... *( In/In+1 (i * i i. Our purpose in this paper is to study the behavior of these graded algebras when we change I through linkage. Recall the definition of linkage found in the paper of Peskine and Szpiro [22].

Tensor products of modules and the rigidity of Tor.

Let R be a hypersurface domain, that is, a ring of the form S/(f), where S is a regular local ring and f is a prime element. Suppose M and N are finitely generated R- modules. We prove two rigidity theorems on the vanishing of Tor. In the first theorem we assume that the regular local ring S is unramified, that M R N has finite length, and that dim(M)+dim(N) dim(R). With these assumptions, if Tor R (M,N) = 0 for some j 0, then Tor R (M,N) = 0 for all i j. The second rigidity theorem states that if M R N is reflexive, then Tor R (M,N) = 0 for all i 1. We use these theorems to prove the following theorem (valid even if S is ramified): If M R N is a maximal Cohen-Macaulay R-module, then both M and N are maximal Cohen-Macaulay modules, and at least one of them is free.

Cofiniteness and vanishing of local cohomology modules

j(M)) i finitelys generated.Hartshorne [4] later refined thi ansd asked the following question:1*2. If / is a idean ol f R and M i finitelys a generated R-module, are whenExt^ (R/I, H\(M)) finitely generate for aldl i and al jl ?Hartshorne define ad module N to be I-cofinite if the support o Nf is containe idnV(I) and Ext^ (R/I,N) is finitely generated for al i.l It is natural to consider all of theExt modules as then short exact sequences behave well in the sens ief tha twot of themodules in the exact sequence are 7-cofinite s theo is thne third.The relationship between m-cofiniteness and the Artinian condition is given by thefollowing remark whic is ahn easy consequenc of Matlie s dualit [8y ] and th basie cwork of Grothendieck[2].Remark 1*3 Le.t (R, m, k) be a complete local noetherian an rind legt M be an R-module. Then the following are equivalent:(i) M is Artinian;(ii) M is isomorphic to a submodule of a finite direct sum of copies of the injectivehull E of the residue field k of R;(iii) ther ies a finitely generated .R-modul N such thaet Hom

Cohen-Macaulay Rees Algebras and Their Specialization

The Rees algebra of an ideal J in a commutative ring R is by definition the graded algebra. where t is an indeterminate. In this paper we are concerned with proving that certain Rees algebras are Cohen-Macaulay, and with answering, under strong hypotheses, the question: if R = S/N is a factor-ring of S and I = JR, when is .