Coda

Summary

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.