the differential equation P f = 0, where f is some holomorphic function in C, say. The vector space consisting of the solutions of that equation is given by the space of homomorphisms of D-modules . 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 DX is defined to be the OX-algebra generated by the vector fields on X, interpreted as derivations. A (left) DX-module M is an OX-module with a left action of DX on it. 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 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. As the ring DX 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 ⊗ ΩX, where ΩX 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 := − Liev (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, ..., 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 , where the are regular functions on X. 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