D-module - Knowlpedia

Definition

Let $X$ be a smooth algebraic variety in characteristic $0$ . A left $D$ -module on $X$ is a sheaf $\mathcal M$ of left modules over the sheaf $\mathcal D_X$ of differential operators whose underlying $\mathcal O_X$ -module is quasi-coherent. Right $D$ -modules are defined similarly.

Flat connections

An $\mathcal O_X$ -coherent $D$ -module is equivalently a finite-rank locally free $\mathcal O_X$ -module with an integrable connection. General $D$ -modules need not be coherent over $\mathcal O_X$ ; they allow singularities and distribution-like objects.

Stacks and derived categories

For a suitable algebraic stack $\mathcal X$ , $D$ -modules can be defined by descent from smooth charts. In geometric Langlands, $D\text{-}\operatorname{mod}(\mathcal X)$ usually denotes a stable derived category; its precise descent and functorial conventions must be fixed separately from the variety-level definition above.

References

Alexander Beilinson and Joseph Bernstein, “A proof of Jantzen conjectures,” in I. M. Gelfand Seminar, Advances in Soviet Mathematics 16 (1993), 1–50.
Masaki Kashiwara, “On the maximally overdetermined system of linear differential equations I,” Publ. RIMS 10 (1975), 563–579. DOI.