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).