Bernd Ulrich

What is the Rees algebra of a module

In this paper we show that the Rees algebra can be made into a functor on modules over a ring in a way that extends its classical definition for ideals. The Rees algebra of a module M may be computed in terms of a maximal map f from M to a free module as the image of the map induced by f on symmetric algebras. We show that the analytic spread and reductions of M can be determined from any embedding of M into a free module, and in characteristic 0-but not in positive characteristic!-the Rees algebra itself can be computed from any such embedding.

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.

Liaison and Castelnuovo-Mumford regularity

In this article we establish bounds for the Castelnuovo-Mumford regularity of projective schemes in terms of the degrees of their defining equations. The main new ingredient in our proofs is to show that generic residual intersections of complete intersection rational singularities again have rational singularities. When applied to the theory of residual intersections this circle of ideas also sheds new light on some known classes of free resolutions of residual ideals.

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

Rees algebras of ideals with low codimension

For certain grade two perfect ideals, there is an expected description of the equations of the Rees algebra. In this paper, the Cohen– Macaulayness of the Rees algebra, numerical invariants of the ideal, and a condition on the minors of a presentation matrix of the ideal are shown to be related to the equations having this form.

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

The structure of the core of ideals

Abstract. The core of an R-ideal I is the intersection of all reductions of I. This object was introduced by D. Rees and J. Sally and later studied by C. Huneke and I. Swanson, who showed in particular its connection to J. Lipmans notion of adjoint of an ideal. Being an a priori infinite intersection of ideals, the core is difficult to describe explicitly. We prove in a broad setting that: core(I) is a finite intersection of minimal reductions; core(I) is a finite intersection of general minimal reductions; core(I) is the contraction to R of a ‘universal’ ideal; core(I) behaves well under flat extensions. The proofs are based on general multiplicity estimates for certain modules.

Ideals having the expected reduction number

It is known that the blow-up algebras of an ideal enjoy many good properties if the reduction number of the ideal satisfies an expected upper bound. In this paper we characterize ideals that have this expected reduction number. Applied to perfect ideals of grade two, our result yields a concrete criterion in terms of Fitting ideals for when the Rees algebra is Cohen-Macaulay.

Core and residual intersections of ideals

D. Rees and J. Sally defined the core of an R-ideal I as the in- tersection of all (minimal) reductions of I. However, it is not easy to give an explicit characterization of it in terms of data attached to the ideal. Until recently, the only case in which a closed formula was known is the one of inte- grally closed ideals in a two-dimensional regular local ring, due to C. Huneke and I. Swanson. The main result of this paper explicitly describes the core of a broad class of ideals with good residual properties in an arbitrary local Cohen-Macaulay ring. We also find sharp bounds on the number of minimal reductions that one needs to intersect to get the core.

A formula for the core of an ideal

Abstract.The core of an ideal is the intersection of all its reductions. For large classes of ideals I we explicitly describe the core as a colon ideal of a power of a single reduction and a power of I.