nLab D-module

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Topos Theory

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Higher algebra

higher algebra

universal algebra

Contents

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).

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.

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 $(E_0, E_1,\ldots)$ 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 $X$.

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.

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).

Relation to geometric representation theory

For the moment see at Harish Chandra transform.

References

A comprehensive account is in chapter 2 of

Discussion in derived algebraic geometry is in

Lecture notes include

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

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

• 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

• 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