Aron Simis

On the canonical module of the Rees algebra and the associated graded ring of an ideal

The investigations of this paper originate in the following question: Suppose I is an ideal of a local Gorenstein ring (R, m), whose associated graded ring gr,(R) is CohenMacaulay. Under which extra conditions is gr,(R) actually a Gorenstein ring? Hochster shows in [ 151 that gr,(R) is a Gorenstein ring if R is factorial and gr,(R) is a domain. Hochster’s arguments work as well if one only requires R to be a Gorenstein ring. However, the condition that gr,(R) is a domain cannot be weakened much. In fact, we given an example of a local complete intersection (R, m) whose associated graded ring gr,,,(R) is a reduced Cohen-Macaulay ring, but not a Gorenstein ring. The question when gr,(R) is a Gorenstein ring has a more satisfying answer for ideals I generated by a d-sequence, for which R/I is a Cohen-Macaulay ring. d-sequences were introduced and studied by Huneke. Their relevance in the study of powers of ideals was shown by Huneke in [ 161. For further investigations on d-sequences the reader is referred to [ 111. Now, given an ideal I generated by a d-sequence and such that R/I is CohenMacaulay we show that gr,(R) is a Gorenstein ring if and only if I is strongly Cohen-Macaulay. Again, the notion “strongly Cohen-Macaulay” was introduced by Huneke. It means that the Koszul homology of a system of generators of I is Cohen-Macaulay. Huneke showed in [ 191 that any ideal in the linkage class of a complete intersection is strongly Cohen-Macaulay. This result provides us with plenty of interesting examples. The main subject of the paper, however, is the study of the canonical module of gr,(R) and of the Rees algebra R[If] = R@Zr@12t2@ ... sR[t]. In [14] the canonical class [os] of the Rees algebra S= R[Zr] was determined under the assumption that I is

Rees Algebras of Modules

We study Rees algebras of modules within a fairly general framework. We introduce an approach through the notion of Bourbaki ideals that allows the use of deformation theory. One can talk about the (essentially unique) generic Bourbaki ideal I(E) of a module E which, in many situations, allows one to reduce the nature of the Rees algebra of E to that of its Bourbaki ideal I(E). Properties such as Cohen?Macaulayness, normality and being of linear type are viewed from this perspective. The known numerical invariants, such as the analytic spread, the reduction number and the analytic deviation, of an ideal and its associated algebras are considered in the case of modules. Corresponding notions of complete intersection, almost complete intersection and equimultiple modules are examined in some detail. Special consideration is given to certain modules which are fairly ubiquitous because interesting vector bundles appear in this way. For these modules one is able to estimate the reduction number and other invariants in terms of the Buchsbaum?Rim multiplicity.

Cohen-Macaulay Rees algebras and degrees of polynomial relations*

(E-mail: aron@sunrnp.vfba.br) 2 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA (E-mail: 21144bfu@msu.edu) 3 Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA (E-mail: vasconce@rings.rutgers.edu) Received: 5 October 1993

On the arithmetic and homology of algebras of linear type

Three modifications of the symmetric algebra of a module are introduced and their arithmetical and homological properties studied. Emphasis is placed on converting syzygetic properties of the modules into ideal theoretic properties of the algebras, e.g. Cohen-Macaulayness, factoriality. The main tools are certain Fitting ideals of the module and an extension to modules of a complex of not necessarily free modules that we have used in studying blowing-up rings.

Codimension, multiplicity and integral extensions

Abstract Let AˆB be a homogeneous inclusion of standard graded algebras with A 0 = B 0 .To relate properties of A and B we intermediate with another algebra, the associatedgradedring G =gr A 1 B ( B ).Wegivecriteriaastowhentheextension AˆB isintegralor birational in terms of the codimension of certain modules associated to G .Wealso introduce a series of multiplicities associated to the extension AˆB . There areapplications to the extension of two Rees algebras of modules and to estimating the(ordinary) multiplicity of A in terms of that of B and of related rings. Many earlierresults by several authors are recovered quickly.1. Introduction Let AˆB be a homogeneous inclusion of standard graded Noetherian rings with A 0 = B 0 = R . Our primeval goal is to give a uni ed treatment of criteria for theintegrality (resp. birationality) of the extension AˆB which would include many ofthe earlier results that dealt with special cases of this general set up. On one side, theguiding principle has been to pro t from the intertwining between the gradings bymeans of a third graded algebra, namely, the associated graded ring

On the homology of two-dimensional elimination

We study birational maps with empty base locus defined by almost complete intersection ideals. Birationality is shown to be expressed by the equality of two Chern numbers. We provide a relatively effective method for their calculation in terms of certain Hilbert coefficients. In dimension 2 the structure of the irreducible ideals-always complete intersections by a classical theorem of Serre-leads by a natural approach to the calculation of Sylvester determinants. We introduce a computer-assisted method (with a minimal intervention by the computer) which succeeds, in degree @?5, in producing the full sets of equations of the ideals. In the process, it answers affirmatively some questions raised by Cox [Cox, D.A., 2006. Four conjectures: Two for the moving curve ideal and two for the Bezoutian. In: Proceedings of Commutative Algebra and its Interactions with Algebraic Geometry, CIRM, Luminy, France, May 2006 (available in CD media)].

On Birational Maps and Jacobian Matrices

One is concerned with Cremona-like transformations, i.e., rational maps from ℙn to ℙm that are birational onto the image Y ⊂ ℙm and, moreover, the inverse map from Y to ℙn lifts to ℙm. We establish a handy criterion of birationality in terms of certain syzygies and ranks of appropriate matrices and, moreover, give an effective method to explicitly obtaining the inverse map. A handful of classes of Cremona and Cremona-like transformations follow as applications.

A characteristic-free criterion of birationality

Abstract One develops ab initio the theory of rational/birational maps over reduced, but not necessarily irreducible, projective varieties in arbitrary characteristic. A virtual numerical invariant of a rational map is introduced, called the Jacobian dual rank. It is proved that a rational map in this general setup is birational if and only if the Jacobian dual rank is well defined and attains its maximal possible value. Even in the “classical” case where the source variety is irreducible there is some gain for this invariant over the degree of the map because, on one hand, it relates naturally to constructs in commutative algebra and, on the other hand, is effectively computable. Applications are given to results only known so far in characteristic zero. One curious byproduct is an alternative approach to deal with the result of Dolgachev concerning the degree of a plane polar Cremona map.

Constraints for the normality of monomial subrings and birationality

Let k be a field and let F C k[x 1 ,...,x n ] be a finite set of monomials whose exponents lie on a positive hyperplane. We give necessary conditions for the normality of both the Rees algebra R[Ft] and the subring k[F]. If the monomials in F have the same degree, one of the consequences is a criterion for the k-rational map F: P n-1 k --→ P m-1 k defined by F to be birational onto its image.

The module of derivations of a Stanley-Reisner ring

An explicit description is given of the module Der(k[X]/I, k[X]/I) of the derivations of the residue ring k[X]/I, where I is an ideal generated by monomials whose exponents are prime to the characteristic of the field k (this includes the case of square free monomials in any characteristic and the case of arbitrary monomials in characteristic zero). In the case where I is generated by square free monomials, this description is interpreted in terms of the corresponding abstract simplicial complex A. Sharp bounds for the depth of this module are obtained in terms of the depths of the face rings of certain subcomplexes Ai related to the stars of the vertices vi of A. The case of a Cohen-Macaulay simplicial complex A is discussed in some detail: it is shown that Der(k[A], k[A]) is a Cohen-Macaulay module if and only if depthAi > dim A 1 for every vertex vi . A measure of triviality of the complexes Ai is introduced in terms of certain star corners of vi . A curious corollary of the main structural result is an affirmative answer in the present context to the conjecture of Herzog-Vasconcelos on the finite projective dimension of the k[X]/I-module Der(k[X]/I, k[X]/I).