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.), which is quasicoherent as $O_X$-module. 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 $\mathcal{D}$-modules on a smooth scheme $X$ may equivalently be identified with the category of quasicoherent sheaves on its deRham space $dR(X)$ (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 $\Pi_{inf}(X)$ of $X$;

• a quasicoherent sheaf on $dR(X)$ 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 $\infty$-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.

Positive characteristic

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 $C^\infty$ category, which might explain my puzzlement.

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 $\text{Der}_X(O_X,O_X)$. Indeed, if $X$ is smooth over a base field $k$, the ring of Grothendieck differential operators $D_X$ will not be Noetherian, instead being generated by operators like $\frac{1}{(p^k)!} (\frac{d}{dt})^{p^k}$. Thus an $O_X$-coherent $D_X$-module will never be coherent over $D_X$.

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 $(E_0, E_1,\ldots)$ of locally free $O_X$-modules together with $O_X$-isomorphisms $\sigma_i: E_i\rightarrow F^* E_{i+1}$, where $F$ is the Frobenius endomorphism of $X$ Gieseker ‘75, Theorem 1.3.

In contrast, modules over the ring of crystalline differential operators are tautologically $O_X$-modules equipped with an integrable connection. These have a different flavor than in characteristic zero because of the existence of $p$-curvature, or equivalently, because the ring of crystalline differential operators is an Azumaya algebra.

Properties

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).

The most efficient and intuitive way to define the six operations on D-modules is to transfer them from Ω-modules? (i.e., modules over the differential graded algebra of differential forms) using Koszul duality. The six operations on Ω-modules? can be defined in the standard way using the fact that differential forms can be pulled back, unlike differential operators. See the article Koszul duality for more information.

Relation to geometric representation theory

For the moment see at Harish Chandra transform.

Related concepts

• coherent D-module, holonomic D-module

• arithmetic D-module

• linear differential equation

• D-geometry

• D-scheme

• D-algebra

• Riemann-Hilbert correspondence

• crystalline cohomology

• Weyl algebra, regular differential operator, local system, differential bimodule, Grothendieck connection, crystal, algebraic analysis.

• Hecke category, Harish Chandra transform

References

A comprehensive account is in chapter 2 of

• Alexander Beilinson and Vladimir Drinfeld, chapter 2 of Chiral Algebras

• Armand Borel et al., Algebraic D-modules, Perspectives in Mathematics, Academic Press, 1987 (djvu)

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

Discussion in derived algebraic geometry is in

• Dennis Gaitsgory, Nick Rozenblyum, Crystals and D-modules, Pure and Applied Mathematics Quarterly Volume 10 (2014) Number 1 (arXiv:1111.2087, publisher)

Lecture notes:

• Jacob Lurie, Notes on crystals and algebraic $\mathcal{D}$-modules (2010) [pdf]

• Severino C. Coutinho, A primer of algebraic D-modules, London Math. Soc. Stud. Texts 33, Cambridge University Press (1995) [doi:10.1017/CBO9780511623653]

• Joseph Bernstein, Algebraic theory of D-modules (pdf, 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

• 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

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

• Pavel Etingof, Formalism of six functors on all (coherent) D-modules (pdf)

• David Ben-Zvi, David Nadler, section 3 of The Character Theory of a Complex Group (arXiv:0904.1247)

See also

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

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

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