I presented this paper at the American Mathematical Society Meeting in Lincoln, NE in October 2011. Here you can find the slides of my talk.
Introduction. In this paper, we consider the following question, attributed to Vasconcelos. Given an ideal I in a regular local ring R, if we know that both R/I and R/I^2 are Cohen-Macaulay can we conclude that R/I is Gorenstein?
In the literature there are several instances where Vasconcelos Question is proved to be true. Most remarkably, in the cases of licci ideals (that includes Cohen-Macaulay ideals of codimension two) and squarefree monomial ideals.
Quick Summary. In our first paper with Y. Xie, we investigate this question of Vasconcelos. We provide several classes of ideals for which the question has a positive answer. However, we exhibit examples proving at once that the answer to Vasconcelos' Question in general is negative and the sharpness of our main result. several crucial ideas of this paper were developed during the Mathematics Research Communities (MRC) Week in the wonderful location of Snowbird, UT. We are indebted with C. Huneke, C. Polini and B. Ulrich for several useful conversations and insightful remarks.
Here you can also find slides on this paper that I presented at the AMS meeting, in October 2011.
Introduction.A celebrated inequality proved by Abhyankar states that if R is a local Cohen-Macaulay ring R, then its multiplicity, e(R), is at least ecodim(R)+1, where ecodim(R) is the embedding codimension of R. If equality is attached, R is said to have minimal multiplicity. If e(R)=ecodim(R)+2, one says that R has almost minimal multiplicity.
Sally in the early 80's conjectured, that, if R has almost minimal multiplicity, then the associated graded ring is almost Cohen-Macaulay. The Conjecture was proved only 13 years later, independently, by Rossi-Valla and Wang. Several generalizations were attempted afterwards. The most far-reaching, was proved by Rossi-Valla in 2001, who introduced the notion of stretched m-primray ideals and used it to prove that the associated graded rings of these ideals are almost Cohen-Macaulay (under some additional assumption).
Quick Summary. In this second joint paper with Y. Xie, we provide a unifying approach to the study of the (almost) Cohen-Macaulayness of associated graded rings of ideals of arbitrary height. Our main theorems generalize at once, for instance, results of Sally, Rossi-Valla,Wang, Elias, Rossi, Corso-Polini-Vaz Pinto, Huckaba and Polini-Xie.
In fact, we introduce j-stretched ideals, which can be thought as the higher dimensional version of stretched ideals. The `j' in the name comes from the fact that this ideals generalize ideals having minimal and almost minimal j-multiplicity.
In contrast to the classical definition of stretchedness, which requires the ideal to be 0-dimensional, our definition works for ideals of arbitrary dimension. Also, we prove that when both notions are defined, j-stretched ideals are more general than stretched ideals.
One of our main results states that, if I is j-stretched, then G is Cohen-Macaulay if and only if the reduction number of I equals the index of nilpotency of I.
Our second main result proves a generalized version of Sally's Conjecture, that holds for j-stretched ideals. It generalizes both the m-primary versions of this result and the main results of a recent paper of Polini-Xie.
Along the way, we answer a question of Sally asking how much does the property of being `stretched' depend on the choice of the minimal reduction.
Introduction. Two ideals I and J of same codimension (height) c are linked if there exists a complete intersection ideal G contained in both I and J such that G : I=J and G : J=I. (Note: in the above definition one could only ask G to be Gorenstein. In this case I and J are said to be Gorenstein linked (or G-linked). There are several very interesting results in Gorenstein linkage, however, for the purpose of this paper, we restrict our attention to the case where G has to be a complete intersection)
Linkage theory has been successfully used in several, very different contexts (classification of varities, production of classes of ideals satisfying strong or unusual homological properties, study of Hilbert schemes, etc.) by many authors, such as R. Hartshorne, C. Huneke, R. Lazarsfeld, J. C. Migliore, A. P. Rao, B. Ulrich, just to name a few (but there are so many more!!). A large body of the literature on linkage has addressed questions relative to the most relevant (and wellbehaved) class of ideals in linkage: licci ideals, that is, ideals linked in finitely many steps to an ideal generated by a regular sequence (that is, a complete intersection ideal). However, for a non-licci ideal I, very little is known about the structure of its linkage class (that is, the colletion of the ideals that can be linked to I in finitely many steps).
Quick Summary. In the present paper, we provide a new approach to study the (even) linkage classes of non licci ideals. We then use it to provide new evidence towards some long-standing conjectures (e.g. Buchsbaum-Eisenbud-Horrocks Conjecture).
More specifically, we introduce a general notion of minimal representatives for any given (even) linkage class. We show tthat these ideals essentially play the same role in the linkage class of I that complete intersection ideals play for licci ideals, have the smallest possible Betti numbers and the best homological properties in the even linkage class of I. Also, since a priori it is not obvious that these elements exist, we prove, under reasonable assumptions on the ideal (or its even linkage class), that these minimal elements exist and are even unique (up to some equivalence).
We then exhibit several classes of ideals that are the minimal representatives of their even linkage class. The following ideals are minimal, provided they are not licci:
1- powers of maximal ideals;
2- powers of complete intersection ideals;
3- ideals of fat points in P^n (n>3);
(the cases 1-3 generalize results of work of Polini-Ulrich)
4- rigid isolated singularities;
5- homogeneous ideals having a linear graded free resolution;
6- Any ideal generated by the t x t minors of a r x s matrix with non-unit entries, if it has the expected height;
7- Any ideal generated by the t x t Pfaffians of a r x s skew-symmetric matrix with non-unit entries, if it has the expected height ;
8- ideals with almost minimal multiplicity having the smallest possible embedding codimension in their even linkage class;
The theory developed in this paper allows us to produce new evidence towards long-standing conjectures. For instance, any ideal in the even linkage class of any ideal I that falls in one of the categories 1-8 above must satisfy the Buchsbaum-Eisenbud-Horrocks Conjecture.
In some of the above cases, this was already known. However, this application gives a unifying approach and shows how to use these minimal elements to obtain information about the whole even linkage class.
