Chapter 10 - Holonomic modules

Summary

The most important A_n-modules are the holonomic modules, also known among PDE theorists as maximally overdetermined systems. An A_n-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 A_n-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 A_n-module. We already know an example of a holonomic A_n- module, viz. K[X] = K[x₁, …, x_n]. We also know that A_n 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 A_n-modules are holonomic.

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

Proof: (1) These follow from Bernstein's inequality. Let M be a left A_n 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).