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25816 results in Abstract analysis

Page 957 of 1291

  • Chapter 5 - Modules over the Weyl algebra

  • S. C. Coutinho
  • Book:
    A Primer of Algebraic D-Modules
    Published online:
    29 December 2009
    Print publication:
    29 May 1995, pp 36-43

  • Summary

    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.

  • Chapter 10 - Holonomic modules

  • S. C. Coutinho
  • Book:
    A Primer of Algebraic D-Modules
    Published online:
    29 December 2009
    Print publication:
    29 May 1995, pp 86-96

  • Summary

    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).

Page 957 of 1291