S. C. Coutinho

D-SIMPLE RINGS AND SIMPLE D-MODULES

We use certain derivations of a polynomial ring, that do not leave any proper nonzero ideal invariant, to construct simple non-holonomic modules over the nth Weyl algebra. This approach extends to the rings of differential operators of other smooth afne varieties, like smooth quadric surfaces.

Algebraic solutions of holomorphic foliations: An algorithmic approach

We present two algorithms that can be used to check whether a given holomorphic foliation of the projective plane has an algebraic solution, and discuss the performance of their implementations in the computer algebra system Singular.

On the differential simplicity of polynomial rings

Abstract Commutative differentially simple rings have proved to be quite useful as a source of examples in noncommutative algebra. In this paper we use the theory of holomorphic foliations to construct new families of derivations with respect to which the polynomial ring over a field of characteristic zero is differentially simple.

The Many Avatars of a Simple Algebra

auspices of deformation theory. In this paper we survey the incarnations of the Weyl algebra associated to several formalisms of quantum mechanics. Beginning with the moment of conception in the 1920s, we work our way through matrix mechanics, Schrodingers equation and Diracs formalism. After a brief interlude where rings of differential operators are introduced, we return to quantum theory to look at quantisation by deformation and its version of the Weyl algebra.

NONHOLONOMIC SIMPLE D -MODULES FROM SIMPLE DERIVATIONS

We give new examples of affine sufaces whose rings of coordinates are d-simple and use these examples to construct simple nonholonomic Dmodules over these surfaces.

Conormal varieties and characteristic varieties

We show that the conormal variety of a quasihomogeneous hypersurface in Cn, for n ≥ 4, whose link is a Q-homology sphere is not the characteristic variety of any D-module.

Foliations of multiprojective spaces and a conjecture of Bernstein and Lunts

We use foliations of multiprojective spaces defined by Hamiltonian functions on the underlying affine space to prove the three dimensional case of a conjecture of Bernstein and Lunts, according to which the symbol of a generic first-order differential operator gives rise to a hypersurface of the cotangent bundle which does not contain involutive conical subvarieties apart from the zero section and fibres of the bundle.

The Quest for Quotient Rings (Of Noncommutative Noetherian Rings)

Articles on the history of mathematics can be written from many different perspectives. Some aim to survey a more or less wide landscape, and require the observer to watch from afar as theories develop and movements are born or become obsolete. At the other extreme, there are those that try to shed light on the history of particular theorems and on the people who created them. This article belongs to this second category. It is an attempt to explain Goldie’s theorems on quotient rings in the context of the life and times of the man who discovered them.

On the differential simplicity of affine rings

We prove that every complex regular affine ring is differentially simple relative to a set with only two derivations. The study of the differential simplicity of commutative rings has known a resurgence of interest in recent years, but its basic results go back at least to the 1950s. In order to review the results that seem most relevant to the theme of this note we introduce a few definitions. Let K be a commutative ring. A derivation d of a (commutative) K-algebra R is an endomorphism of the additive group of R that satisfies d(K) = 0 and Leibniz’s rule for the differentiation of a product, namely d(ab) = ad(b) + bd(a) for all a, b ∈ R. Denoting by DerK(R) the set of all derivations of R, let ∅ 6= D ⊂ DerK(R) be a family (finite or not) of derivations of R. An ideal I of R is D-stable if d(I) ⊂ I for all d ∈ D. Of course the ideals 0 and R are always D-stable. If R has no other D-stable ideal it is called D-simple (or differentially simple if D = DerK(R)). The 1960s saw a flurry of results on differentially simple rings, among them the classification of differentially simple algebras that are finite dimensional over a field [2] or that are affine over an algebraically closed field of positive characteristic [9]. From our point of view the most interesting result of that decade was Seidenberg’s proof in [14, Theorem 3, p. 26] that every regular affine ring over a field of characteristic zero is differentially simple. In the 1970s some of the most impressive results in the area concerned rings that are differentially simple relative to a one element family {d}. To simplify the notation these rings are called d-simple and the corresponding derivation d is said to be simple. In [10] R. Hart proved that the local ring at a nonsingular point of an irreducible variety over a field of characteristic zero is always d-simple. In the same paper he exhibited an example of a regular affine Q-algebra R that is not d-simple, for any choice of d ∈ DerK(R). On the other hand, George Bergman (unpublished) showed that the ring of polynomials Q[x, y] is d-simple for an appropriately chosen derivation d. More examples of simple derivations have appeared since then in [1], [13], [5], [3] and [6]. For some recent applications in noncommutative algebra see [4], [11] and [8]. Date: November 28, 2011. 1991 Mathematics Subject Classification. Primary: 37F75, 16S32 ; Secondary: 37J30, 32C38, 32S65.

On Homogenous Minimal Involutive Varieties

Let S(2n, k) be the variety of homogeneous polynomials of degree k in 2n variables. The authors of this paper give a computer-assisted