nLab
D-module

Definition

In terms of differential operators

A D-module (introduced by Mikio Sato) is a sheaf of modules over the sheaf $D_X$ of regular differential operators on a ‘variety’ $X$ (the latter notion depends on whether we work over a scheme, manifold, analytic complex manifold etc.). As $O_X$ is a subsheaf of $D_X$ consisting of the zeroth-order differential operators (multiplications by the sections of structure sheaf), every $D_X$-module is an $O_X$-module. Moreover, the (quasi)coherence of $D_X$-modules implies the (quasi)coherence of a $D_X$-module regarded as an $O_X$-module (but not vice versa).

In terms of sheaves on the deRham space

The category of D-modules on a smooth scheme $X$ may equivalently be identified with the category of quasicoherent sheaves on its deRham space $\mathrm{dR}(X)$ (in non-smooth case one needs to work in derived setting, with de Rham stack instead).

Remembering, from this discussion there, that

• the deRham space is the decategorification of the infinitesimal path groupoid of $X$;

• a quasicoherent sheaf on $\mathrm{dR}(X)$ is a generalization of a vector bundle on $X$;

• a vector bundle with a flat connection is an equivariant vector bundle on the infinitesimal path $\mathrm{\infty }$-groupoid ${\Pi }^{\inf }$ of $X$

this shows pretty manifestly how D-modules are “sheaves of modules with flat connection”, as described more below.

Meaning and usage

$D$-modules are useful as a means of applying the methods of homological algebra and sheaf theory to the study of analytic systems of partial differential equations.

Insofar as an $O$-module on a ringed site $(X,O)$ can be interpreted as a generalization of the sheaf of sections of a vector bundle on $X$, a D$-$module can be interpreted as a generalization of the sheaf of sections of a vector bundle on $X$ with flat connection $\nabla$. The idea is that the action of the differential operation given by a vector field $v$ on $X$ on a section $\sigma$ of the sheaf (over some patch $U$) is to be thought of as the covariant derivative $\sigma \mapsto {\nabla }_v\sigma$ with respect to the flat connection $\nabla$.

In fact when $X$ is a complex analytic manifold, any $D_X$-module which is coherent as $O_X$-module is isomorphic to the sheaf of sections of some holomorphic vector bundle with flat connection. Furthermore, the subcategory of nonsingular $D_X$-modules coherent as $D_X$-modules is equivalent to the category of local systems.

If $X$ is a variety over a field of positive characteristic $p$, the terms ”$O_X$-coherent coherent $D_X$-module” and “vector bundle with flat connection” are not interchangeable, since $D_X$ no longer is the enveloping algebra of $O_X$ and ${\text{Der}}_X(O_X,O_X)$. A theorem by Katz states that for smooth $X$ the category of $O_X$-coherent $D_X$-modules is equivalent to the category with objects sequences $(E_0,E_1,\dots )$ of locally free $O_X$-modules together with $O_X$-isomorphisms ${\sigma }_i:E_i\to F^{*}{E}_{i+1}$, where $F$ is the Frobenius endomorphism of $X$.

John Baez: it would be nice to have a little more explanation about how not every $D$-module that is coherent as an $O$-module is coherent as a $D$-module. If I understand correctly, this may be the same question as how not every holomorphic vector bundle with flat connection is a local system. Perhaps the answer can be found under local system? Apparently not. Perhaps the point is that not every flat connection on a holomorphic vector bundle is locally holomorphically trivializable? If so, this is different than how it works in the $C^{\mathrm{\infty }}$ category, which might explain my puzzlement.

References and related entries

Related $n$lab entries include Weyl algebra, regular differential operator, local system, differential bimodule, Grothendieck connection, crystal, algebraic analysis.

• A. Beilinson, I. N. Bernstein, A proof of Jantzen conjecture, Adv. in Soviet Math. 16, Part 1 (1993), 1-50. MR95a:22022

• S. C. Coutinho, A primer of algebraic D-modules, London Math. Soc. Stud. Texts, 33, Cambridge University Press, Cambridge, 1995. xii+207 pp.

• Notes on D-modules: Joseph Bernstein’s notes, Peter Schneiders’ notes, Dragan Miličić’s notes

• D. Gieseker, Flat vector bundles and the fundamental group in non-zero characteristics, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 1, 1–31.

• R. Bezrukavnikov, MIT course notes, pdf

• Notes in Gaitsgory’s seminar pdf

• A. Beĭlinson, J. Bernstein, A proof of Jantzen’s conjectures, I. M. Gelʹfand Seminar, 1–50, Adv. Soviet Math., 16, Part 1, Amer. Math. Soc. 1993, pdf

• R. Hotta, K. Takeuchi, T. Tanisaki, D-modules, perverse sheaves, and representation theory, Progress in Mathematics 236, Birkhäuser

• A. Borel et al., Algebraic D-modules, Perspectives in Mathematics, Academic Press, 1987.

• J.-E. Björk, Rings of differential operators, North-Holland Math. Library 21. North-Holland Publ. 1979. xvii+374 pp.

• M. Kashiwara, W.Schmid, Quasi-equivariant D-modules, equivariant derived category, and representations of reductive Lie groups, Lie Theory and Geometry, in Honor of Bertram Kostant, Progress in Mathematics, Birkhäuser, 1994, pp. 457–488

• P. Maisonobe, C. Sabbah, D-modules cohérents et holonomes, Hermann, Paris 1993.

Blog discussion

• Secret Blogging Seminar Musings on D-modules, Musings on D-modules, part 2

• The Everything Seminar D-module Basics I, D-Module Basics II.