Topos Theory

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In terms of differential operators

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

In terms of sheaves on the deRham space

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

(Lurie, above theorem 0.4, Gaitsgory-Rozenblyum 11, 2.1.1)

Remembering, from this discussion there, that

  • the deRham space is the decategorification of the infinitesimal path groupoid Π inf(X)\Pi_{inf}(X) of XX;

  • a quasicoherent sheaf on dR(X)dR(X) is a generalization of a vector bundle on XX;

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

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

Meaning and usage

DD-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 OO-module on a ringed site (X,O)(X, O) can be interpreted as a generalization of the sheaf of sections of a vector bundle on XX, a D-module can be interpreted as a generalization of the sheaf of sections of a vector bundle on XX with flat connection \nabla. The idea is that the action of the differential operation given by a vector field vv on XX on a section σ\sigma of the sheaf (over some patch UU) is to be thought of as the covariant derivative σ vσ\sigma \mapsto \nabla_v \sigma with respect to the flat connection \nabla.

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

If XX is a variety over a field of positive characteristic pp, the terms “O XO_X-coherent coherent D XD_X-module” and “vector bundle with flat connection” are not interchangeable, since D XD_X no longer is the enveloping algebra of O XO_X and Der X(O X,O X)\text{Der}_X(O_X,O_X). A theorem by Katz states that for smooth XX the category of O XO_X-coherent D XD_X-modules is equivalent to the category with objects sequences (E 0,E 1,)(E_0, E_1,\ldots) of locally free O XO_X-modules together with O XO_X-isomorphisms σ i:E iF *E i+1\sigma_i: E_i\rightarrow F^* E_{i+1}, where FF is the Frobenius endomorphism of XX.

John Baez: it would be nice to have a little more explanation about how not every DD-module that is coherent as an OO-module is coherent as a DD-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 C^\infty category, which might explain my puzzlement.


Six operations yoga

Discussion of six operations yoga for pull-push of (coherent, holonomic) D-modules is in (Bernstein, around p. 18). This is reviewed for instance in (Etingof, Ben-Zvi & Nadler 09).

Relation to geometric representation theory

For the moment see at Harish Chandra transform.


A comprehensive account is in chapter 2 of

Discussion in derived algebraic geometry is in

Lecture notes include

See also

  • Morihiko Saito, Induced D-modules and differential complexes, Bull. Soc. Math. France 117 (1989), 361–387, 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.

  • 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

  • M. Kashiwara, D-modules and representation theory of Lie groups,

    Annales de l’institut Fourier, 43 no. 5 (1993), p. 1597-1618, article, MR95b:22033

  • P. Maisonobe, C. Sabbah, D-modules cohérents et holonomes, Travaux en cours, Hermann, Paris 1993. (collection of lecture notes)

  • Donu Arapura, Notes on D-modules and connection with Hodge theory, pdf

  • 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

Review of six operations yoga for D-modules is in

See also

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

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Last revised on November 7, 2017 at 13:09:08. See the history of this page for a list of all contributions to it.