D-module - D-modules On Algebraic Varieties

D-modules On Algebraic Varieties

The general theory of D-modules is developed on a smooth algebraic variety X defined over an algebraically closed field K of characteristic zero, such as K = C. The sheaf of differential operators D_X is defined to be the O_X-module generated by the vector fields on X, interpreted as derivations. A (left) D_X-module M is an O_X-module with a left action of D_X. Giving such an action is equivalent to specifying a K-linear map

satisfying

(Leibniz rule)

Here f is a regular function on X, v and w are vector fields, m a local section of M, denotes the commutator. Therefore, if M is in addition a locally free O_X-module, giving M a D-module structure is nothing else than equipping the vector bundle associated to M with a flat (or integrable) connection.

As the ring D_X is noncommutative, left and right D-modules have to be distinguished. However, the two notions can be exchanged, since there is an equivalence of categories between both types of modules, given by mapping a left module M to the tensor product M ⊗ Ω_X, where Ω_X is the line bundle given by the highest exterior power of differential 1-forms on X. This bundle has a natural right action determined by

ω ⋅ v := − Lie_v (ω),

where v is a differential operator of order one, that is to say a vector field, ω a n-form (n = dim X), and Lie denotes the Lie derivative.

Locally, after choosing some system of coordinates x₁, ..., x_n (n = dim X) on X, which determine a basis ∂₁, ..., ∂_n of the tangent space of X, sections of D_X can be uniquely represented as expressions

, where the are regular functions on X.

In particular, when X is the n-dimensional affine space, this D_X is the Weyl algebra in n variables.

Many basic properties of D-modules are local and parallel the situation of coherent sheaves. This builds on the fact that D_X is a locally free sheaf of O_X-modules, albeit of infinite rank, as the above-mentioned O_X-basis shows. A D_X-module that is coherent as an O_X-module can be shown to be necessarily locally free (of finite rank).