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.

This chapter collects a number of important examples of modules over the Weyl algebra. The prototype of all the examples we discuss here is the polynomial ring in n variables; and with it we shall begin. The reader is expected to be familiar with the basic notions of module theory, as explained in [Cohn, Ch.10].

THE POLYNOMIAL RING.

In Ch. 1, the Weyl algebra was constructed as a subring of an endomorphism ring. Writing K[X] for the polynomial ring K[x1, …, xn] we have that An(K) is a subring of EndKK[X]. One deduces from this that the polynomial ring is a left An-module. Thus the action of xi on K[X] is by straightforward multiplication; whilst ∂i acts by differentiation with respect to xn. This is a very important example, and we shall study it in some detail. Let us first recall some basic definitions.

Let us first recall some basic definitions. Let R be a ring. An R-module is irreducible, or simple, if it has no proper submodules. Let M be a left R-module. An element u ∈ M is a torsion element if annR(u) is a non-zero left ideal. If every element of M is torsion, then M is called a torsion module.

The most important An-modules are the holonomic modules, also known among PDE theorists as maximally overdetermined systems. An An-module is holonomic if it has dimension n. Ordinary differential equations with polynomial coefficients correspond to holonomic modules. In this chapter we begin the study of holonomic modules, which will be one of the central topics of the second half of the book.

DEFINITION AND EXAMPLES.

A finitely generated left An-module is holonomic if it is zero, or if it has dimension n. Recall that by Bernstein's inequality this is the minimal possible dimension for a non-zero An-module. We already know an example of a holonomic An- module, viz. K[X] = K[x1, …, xn]. We also know that An itself is not a holonomic module: it has dimension 2n.

It is easy to construct holonomic modules if n = 1. Let I ≠ 0 be a left ideal of A1. By Corollary 9.3.5, d(A1/I) ≤ 1. If I ≠ A1 then, by Bernstein's inequality, d(A1/I) = 1. Hence A1/I is a holonomic A1-module.

This is wonderful source of examples, which will pour forth with the help of the next proposition.

Proposition. Let n be a positive integer.

1. Submodules and quotients of holonomic An-modules are holonomic.

2. Finite sums of holonomic An-modules are holonomic.

Proof: (1) These follow from Bernstein's inequality. Let M be a left An module, and N a submodule of M. From Theorem 9.3.2, d(N) ≤ d(M) and d(M/N) ≤ d(M). Since d(M) = n, and using Bernstein's inequality, we deduce that d(N) = d(M/N) are also equal to n. Thus N and M/N are holonomic. Now (2) follows from Corollary 9.3.3 and (1).