D-module

In mathematics, a D-module is a module over a ring D of differential operators. The major interest of such D-modules is as an approach to the theory of linear partial differential equations. Since around 1970, D-module theory has been built up, mainly as a response to the ideas of Mikio Sato on algebraic analysis, and expanding on the work of Sato and Joseph Bernstein on the Bernstein–Sato polynomial. Early major results were the Kashiwara constructibility theorem and Kashiwara index theorem of Masaki Kashiwara. The methods of D-module theory have always been drawn from sheaf theory and other techniques with inspiration from the work of Alexander Grothendieck in algebraic geometry. The approach is global in character, and differs from the functional analysis techniques traditionally used to study differential operators. The strongest results are obtained for over-determined systems (holonomic systems), and on the characteristic variety cut out by the symbols, which in the good case is a Lagrangian submanifold of the cotangent bundle of maximal dimension (involutive systems). The techniques were taken up from the side of the Grothendieck school by Zoghman Mebkhout, who obtained a general, version of the Riemann–Hilbert correspondence in all dimensions. The first case of algebraic D-modules are modules over the Weyl algebra An(K) over a field K of characteristic zero. It is the algebra consisting of polynomials in the following variables x1, ..., xn, ∂1, ..., ∂n. where the variables xi and ∂j separately commute with each other, and xi and ∂j commute for i ≠ j, but the commutator satisfies the relation [∂i, xi] = ∂ixi − xi∂i = 1. For any polynomial f(x1, ..., xn), this implies the relation [∂i, f] = ∂f / ∂xi, thereby relating the Weyl algebra to differential equations. An (algebraic) D-module is, by definition, a left module over the ring An(K). Examples for D-modules include the Weyl algebra itself (acting on itself by left multiplication), the (commutative) polynomial ring K[x1, ...