symmetric monoidal (∞,1)-category of spectra
derived smooth geometry
A D-module (introduced by Mikio Sato) is a sheaf of modules over the sheaf of regular differential operators on a ‘variety’ (the latter notion depends on whether we work over a scheme, manifold, analytic complex manifold etc.), which is quasicoherent as -module. As is a subsheaf of consisting of the zeroth-order differential operators (multiplications by the sections of structure sheaf), every -module is an -module. Moreover, the (quasi)coherence of -modules implies the (quasi)coherence of a -module regarded as an -module (but not vice versa).
The category of D-modules on a smooth scheme may equivalently be identified with the category of quasicoherent sheaves on its deRham space (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 ;
a quasicoherent sheaf on is a generalization of a vector bundle on ;
this shows pretty manifestly how D-modules are “sheaves of modules with flat connection”, as described more below.
Insofar as an -module on a ringed site can be interpreted as a generalization of the sheaf of sections of a vector bundle on , a Dmodule can be interpreted as a generalization of the sheaf of sections of a vector bundle on with flat connection . The idea is that the action of the differential operation given by a vector field on on a section of the sheaf (over some patch ) is to be thought of as the covariant derivative with respect to the flat connection .
In fact when is a complex analytic manifold, any -module which is coherent as -module is isomorphic to the sheaf of sections of some holomorphic vector bundle with flat connection. Furthermore, the subcategory of nonsingular -modules coherent as -modules is equivalent to the category of local systems.
If is a variety over a field of positive characteristic , the terms ”-coherent coherent -module” and “vector bundle with flat connection” are not interchangeable, since no longer is the enveloping algebra of and . A theorem by Katz states that for smooth the category of -coherent -modules is equivalent to the category with objects sequences of locally free -modules together with -isomorphisms , where is the Frobenius endomorphism of .
John Baez: it would be nice to have a little more explanation about how not every -module that is coherent as an -module is coherent as a -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 category, which might explain my puzzlement.
A comprehensive account is in chapter 2 of
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 ps, dvi, Peter Schneiders’ notes, Dragan Miličić’s notes, , Localization and representation theory of reductive Lie groups; Victor Ginzburg’s 1998 Chicago notes pdf; A. Braverman-T. Chmutova, Lectures on algebraic D-modules, pdf
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
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, Travaux en cours, Hermann, Paris 1993. (collection of lecture notes)
Nero Budur, On the V-filtration of D-modules, math.AG/0409123, in “Geometric methods in algebra and number theory” Proc. 2003 conf. Univ. of Miami, edited by F. Bogomolov, Yu. Tschinkel