In this chapter we show that the Weyl algebras are members of the family of rings of differential operators. These rings come up in many areas of mathematics: representation theory of Lie algebras, singularity theory and differential equations are some of them.

DEFINITIONS.

Let R be a commutative K-algebra. The ring of differential operators of R is defined, inductively, as a subring of EndK(R). As in the case of the Weyl algebra, we will identify an element a ∈ R with the operator of EndK(R) defined by the rule r ↦ ar, for every r ∈ R.

We now define, inductively, the order of an operator. An operator P ∈ EndK(R) has order zero if [a, P] = 0, for every a ∈ R. Suppose we have defined operators of order < n. An operator P ∈ EndK(R) has order n if it does not have order less than n and [a, P] has order less than n for every a ∈ R. Let Dn(R) denote the set of all operators of EndK(R) of order ≤ n. It is easy to check, from the definitions, that Dn(R) is a K-vector space.

We may characterize the operators of order ≤ 1 in terms of well-known concepts. A derivation of the K-algebra R is a linear operator D of which satisfies Leibniz's rule: D(ab) = aD(b) + bD(a) for every a,b ∈ R. Let DerK(R) denote the K-vector space of all derivations of R. Of course DerK(R) ⊆ EndK(R).

We have seen in the previous chapters that holonomic modules are preserved by inverse images under projections and by direct images under embeddings. However, as we also saw, inverse images under embeddings and direct images under projections do not preserve the fact that a module is finitely generated. Fortunately, though, holonomic modules are preserved by all kinds of inverse and direct images. The proof of this result will use all the machinery that we have developed so far. It gives yet one more way to construct examples of holonomic modules. We retain the notations of 14.1.2.

INVERSE IMAGES

The key to the results in this chapter is a decomposition of polynomial maps in terms of embeddings and projections. The idea goes back to A. Grothendieck.

Let F : X→ Y be a polynomial map. We may decompose F as a composition of three polynomial maps: a projection, an embedding and an isomorphism. The maps are the following. The projection is π : X × Y → Y, defined by π(X, Y) = Y. The isomorphism is G : X × Y → X × Y where G(X, Y) = (X, Y+F(X)). Finally, the embedding is i : X → X × Y, defined by i(X) = (X, 0). One can immediately check that F = π · G · i.

Since this book is only a primer, it is convenient to give the interested reader directions for further study. The comments that follow are based on this author's experience and inevitably reflect his tastes.

First of all, the theory of algebraic D-modules is itself a part of algebraic geometry. Thus we must start with an algebraic variety X. If we assume that X is affine, then its algebraic geometric properties are coded by the ring of polynomial functions on X (and its modules). This is a commutative ring, called the ring of coordinates and denoted by O(X). The ring of differential operators D(X) is the ring of differential operators of O(X) as defined in Ch. 3. If the variety is smooth (non-singular) then D(X) is a simple noetherian ring.

To deal with general varieties it is necessary to introduce sheaves. The structure sheaf keeps the same relation to a general variety as the coordinate ring does to an affine variety. From it we may derive the sheaf of rings of differential operators. If the variety is smooth, this is a coherent sheaf of rings. The purpose of D-module theory is the study of the category of coherent sheaves of modules over the sheaf of rings of differential operators of an algebraic variety.

It is plain that a good knowledge of algebraic geometry is essential to make sense of these statements. The standard reference is the first three chapters of [Hartshorne]. One can also find the required sheaf theory in Serre's beautiful “Faisceaux algébriques cohérents”, [Serre]. But a thorough grounding in classical algebraic geometry is necessary before one tackles this paper.

The Jacobian conjecture was proposed by O.H. Keller in 1939. It asks whether a polynomial endomorphism of ℂn whose Jacobian is constant must be invertible. Despite its simple and reasonable statement, the conjecture has not been proved even in the two dimensional case. In this chapter we show that this conjecture would follow if one could prove that every endomorphism of the Weyl algebra is an automorphism. The chapter opens with a discussion of polynomial maps, which will play a central rôle in the second part of the book. We shall return to the Jacobian conjecture in Ch. 19.

POLYNOMIAL MAPS.

Let F : Kn → Km be a map and p a point of Kn. We say that F is polynomial if there exist F1, …, Fm ∈ K[x1, …, xn] such that F(p) = (F1(p), …, Fm(p)). A polynomial map is called an isomorphism or a polynomial isomorphism if it has an inverse which is also a polynomial map. It is not always the case that a bijective polynomial map has an inverse which is also polynomial. For an example where this does not occur see Exercise 5.1. However, if K = ℂ, every invertible polynomial map has a polynomial inverse. This is proved in [Bass, Connell and Wright; Theorem 2.1].