 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"-algebra 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" on it.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, "m" a local section of "M", [−, −] denotes the commutator.