The major interest of such D-modules is as an approach to the theory of linear partial differential equations.
This approach is global in character, and differs from the functional analysis techniques traditionally used to study differential operators.
The techniques were taken up from the side of the Grothendieck school by Zoghman Mebkhout, who obtained a general, derived category version of the Riemann–Hilbert correspondence in all dimensions.
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 DX is defined to be the OX-algebra generated by the vector fields on X, interpreted as derivations.
Giving such an action is equivalent to specifying a K-linear map satisfying Here f is a regular function on X, v and w are vector fields,
Therefore, if M is in addition a locally free OX-module, giving M a D-module structure is nothing else than equipping the vector bundle associated to M with a flat (or integrable) connection.
[2] Locally, after choosing some system of coordinates x1, ..., xn (n = dim X) on X, which determine a basis ∂1, ..., ∂n of the tangent space of X, sections of DX can be uniquely represented as expressions In particular, when X is the n-dimensional affine space, this DX 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 DX is a locally free sheaf of OX-modules, albeit of infinite rank, as the above-mentioned OX-basis shows.
A DX-module that is coherent as an OX-module can be shown to be necessarily locally free (of finite rank).
D-modules on different algebraic varieties are connected by pullback and pushforward functors comparable to the ones for coherent sheaves.
Notably, standard notions from commutative algebra such as Hilbert polynomial, multiplicity and length of modules carry over to D-modules.
Finitely generated D-modules M are endowed with so-called "good" filtrations F∗M, which are ones compatible with F∗An(K), essentially parallel to the situation of the Artin–Rees lemma.
The A1(K)-module M = A1(K)/A1(K)P (see above) is holonomic for any nonzero differential operator P, but a similar claim for higher-dimensional Weyl algebras does not hold.
The Bernstein filtration not being available on DX for general varieties X, the definition is generalized to arbitrary affine smooth varieties X by means of order filtration on DX, defined by the order of differential operators.
Also, M is holonomic if and only if all cohomology groups of the complex Li∗(M) are finite-dimensional K-vector spaces, where i is the closed immersion of any point of X.