Wolmer V. Vasconcelos

Computational methods in commutative algebra and algebraic geometry

This ACM volume deals with tackling problems that can be represented by data structures which are essentially matrices with polynomial entries, mediated by the disciplines of commutative algebra and algebraic geometry. The discoveries stem from an interdisciplinary branch of research which has been growing steadily over the past decade. The author covers a wide range, from showing how to obtain deep heuristics in a computation of a ring, a module or a morphism, to developing means of solving nonlinear systems of equations - highlighting the use of advanced techniques to bring down the cost of computation. Although intended for advanced students and researchers with interests both in algebra and computation, many parts may be read by anyone with a basic abstract algebra course.

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).

On finitely generated flat modules

Introduction. The aim of this paper is to study conditions which reflect the projectivity of a given finitely generated flat module over a commutative ring. The use of the invariant factors of a module (see below for definition) are very appropriate here: By translating the description by Bourbaki [4] of finitely generated projective modules, one can state that projectivity=flatness+finitely generated invariant factors. Since the invariant factors of a flat module are very peculiar (locally they are either (1) or (0)), the presence of almost any other condition on the module precipitates their finite generation. For instance, consider the following two statements. Let M be a finitely generated flat moGule over the commutative ring R: (i) Let S be a multiplicative set in R consisting of nonzero divisors such that MS (localization of M with respect to S) is Rs-projective; then M is projective. (ii) Let J be the Jacobson radical of R and assume that MIJM is R/J-projective; then M is projective. The first is a result by Endo [8] who uses homological algebra in the proof, while the second can be viewed as a generalization of the well-known fact that over a local ring a finitely generated flat module is free. Next we apply these ideas for a look at finitely generated flat ideals. Even though they are not always projective (see below for an example of a principal flat ideal which is not projective) it can be shown this to be the case when the flat ideal is a finite intersection of primary ideals. The criterion mentioned above says that for a finitely generated flat ideal, projectivity is the same as having finitely generated annihilator. For rings with the weakened form of coherency that finitely generated ideals have finitely generated annihilators, one can even show that any finitely generated flat submodule of a projective module is projective. When used to study the prime ideals of a ring R of weak dimension one we arrive at the fact that a finitely generated prime ideal is either maximal or generated by an idempotent. It is also shown that if every principal ideal is projective then R is semihereditary; if moreover every cyclic flat module is projective, then R is a direct sum of finitely many Priufer domains.

Remarks on the pole-shifting problem over rings

Given a square n-matrix Fand an n-row matrix G, pole-shifting problems consist in obtaining more or less arbitrary characteristic polynomials for F+ GK, for suitable (“feedback”) matrices K. A review of known facts is given, various partial results are proved, and the case n = 2 is studied in some detail.

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.

On the integral closure of ideals

Abstract:Among the several types of closures of an ideal I that have been defined and studied in the past decades, the integral closure has a central place being one of the earliest and most relevant. Despite this role, it is often a difficult challenge to describe it concretely once the generators of I are known. Our aim in this note is to show that in a broad class of ideals their radicals play a fundamental role in testing for integral closedness, and in case , is still helpful in finding some fresh new elements in . Among the classes of ideals under consideration are: complete intersection ideals of codimension two, generic complete intersection ideals, and generically Gorenstein ideals.

Links of prime ideals

We exhibit the elementary but somewhat surprising property that most direct links of prime ideals in Gorenstein rings are equimultiple ideals. It leads to the construction of a bountiful set of Cohen–Macaulay Rees algebras.