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].

As its title says, this book is only a primer; in particular, you will learn very little ‘grammar’ from it. That is not surprising; to speak the language of algebraic D-modules fluently you must first learn some algebraic geometry and be familiar with derived categories. Both of these are beyond the bounds of an elementary textbook.

But you can expect to know the answers to two basic questions by the time you finish the book: what are D-modules? and why D-modules? It is particularly easy to answer the latter, because D-module theory has many interesting applications. Hardly any area of mathematics has been left untouched by this theory. Those that have been touched range from number theory to mathematical physics.

I have tried to include some real applications, but they are not by any means the ones that have caused the greatest impact from the point of view of mathematics at large. To some, they may even seem a little eccentric. That reflects two facts. First, and most important, this is an elementary book. The most interesting applications (to singularity theory and representations of algebraic groups, for example) are way beyond the bounds of such a book. Second, among the applications that were elementary enough to be presented here, I chose the ones that I like the most.

The pre-requisites have been kept to a minimum. So the book should be accessible to final year undergraduates or first year post-graduates. But I have made no effort to write a book that is ‘purely algebraic’. Such a book might be possible, but it would not be true.

Simple rings are very hard to study because most techniques in ring theory depend on the existence of two-sided ideals. In the case of the Weyl algebra, however, we have a way out. As we saw in Ch. 2, one may define a degree for the elements of the Weyl algebra. Using this degree, we construct a commutative ring, k[x] works as a shadow of An. We may then draw an outline of what An really looks like. This is the best method we have for understanding the structure of An and of its modules.

GRADED RINGS

An important feature of a polynomial ring is that it admits a degree function. We want to generalize and formalize what it means for an algebra to have a degree. This leads to the definition of graded rings. These rings find their justification in algebraic geometry, more precisely in projective algebraic geometry; for details see [Hartshorne, Ch. 1, §2], For the sake of completeness, we define graded rings without assuming commutativity.

Let R be a K-algebra. We say that R is graded if there are K-vector subspaces Ri, i ∈ ℕ, such that

1. (1) R = ⊕i∈ℕRi,

2. (2) Ri · Ri ⊆ Ri+.

The Ri are called the homogeneous components of R. The elements of Ri are the homogeneous elements of degree i. If Ri = 0 when i < 0 then we say that the grading is positive. From now on all graded algebras will have a positive grading unless explicitly stated otherwise.

There is very little that one can say about a general ring and its modules. In practice an interesting structure theory will result either if the ring has a topology (which is compatible with its operations), or if it has finite dimension, or some generalization thereof. As an example of the former, we have the theory of C*-algebras. The latter class includes many important rings: algebras that are finite dimensional over a field, PI rings, artinian rings and noetherian rings. It is the last ones that we now study. In particular, we prove that the Weyl algebra is a noetherian ring.

NOETHERIAN MODULES.

In this book we shall be concerned almost exclusively with finitely generated modules. One easily checks that a homomorphic image of a finitely generated module is finitely generated. However a finitely generated module can have a submodule that is not itself finitely generated. An example is the polynomial ring in infinitely many variables K[x1, x2, …]. Taken as a module over itself this ring is a cyclic left module: it is generated by 1. However, the ideal generated by all the variables x1, x2, … cannot be finitely generated: every finite set of polynomials in K[x1, x2, …] uses up only finitely many of the variables.

We get around this problem with a definition. A left R-module is called noetherian if all its submodules are finitely generated. Examples are easy to come by: vector spaces over K are noetherian K-modules. Every ideal of the polynomial ring in one variable K[x] is a noetherian K[x]-module.

There are several equivalent ways to define noetherianness. We chose the most natural. Here are two more.