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\section*{D-module}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

\hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry}

[[!include higher geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_terms_of_differential_operators}{In terms of differential operators}\dotfill \pageref*{in_terms_of_differential_operators} \linebreak
\noindent\hyperlink{in_terms_of_sheaves_on_the_derham_space}{In terms of sheaves on the deRham space}\dotfill \pageref*{in_terms_of_sheaves_on_the_derham_space} \linebreak
\noindent\hyperlink{meaning_and_usage}{Meaning and usage}\dotfill \pageref*{meaning_and_usage} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{SixOperationsYoga}{Six operations yoga}\dotfill \pageref*{SixOperationsYoga} \linebreak
\noindent\hyperlink{RelationToGeometricRepresentationTheory}{Relation to geometric representation theory}\dotfill \pageref*{RelationToGeometricRepresentationTheory} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{blog_discussion}{Blog discussion}\dotfill \pageref*{blog_discussion} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{in_terms_of_differential_operators}{}\subsubsection*{{In terms of differential operators}}\label{in_terms_of_differential_operators}

A \emph{D-module} (introduced by [[Mikio Sato]]) is a [[abelian sheaf|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).

\hypertarget{in_terms_of_sheaves_on_the_derham_space}{}\subsubsection*{{In terms of sheaves on the deRham space}}\label{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 sheaf|quasicoherent sheaves]] on its [[deRham space]] $dR(X)$ (in non-smooth case one needs to work in derived setting, with de Rham stack instead).

(\hyperlink{LurieCrystal}{Lurie, above theorem 0.4}, \hyperlink{GaitsgoryRozenblyum11}{Gaitsgory-Rozenblyum 11, 2.1.1})

Remembering, from this discussion there, that

\begin{itemize}%
\item the deRham space is the decategorification of the [[infinitesimal shape modality|infinitesimal path groupoid]] $\Pi_{inf}(X)$ of $X$;


\item a quasicoherent sheaf on $dR(X)$ is a generalization of a [[vector bundle]] on $X$;


\item a vector bundle with a flat [[connection on a bundle|connection]] is an [[equivariant cohomology|equivariant]] vector bundle on the infinitesimal path $\infty$-groupoid $\Pi^{inf}$ of $X$



\end{itemize}
this shows pretty manifestly how D-modules are ``sheaves of modules with flat connection'', as described more below.

\hypertarget{meaning_and_usage}{}\subsection*{{Meaning and usage}}\label{meaning_and_usage}

$D$-modules are useful as a means of applying the methods of [[homological algebra]] and [[Categories and Sheaves|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 $X$ \emph{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.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{SixOperationsYoga}{}\subsubsection*{{Six operations yoga}}\label{SixOperationsYoga}

Discussion of [[six operations yoga]] for pull-push of ([[coherent D-module|coherent]], [[holonomic D-module|holonomic]]) D-modules is in (\hyperlink{Bernstein}{Bernstein, around p. 18}). This is reviewed for instance in (\hyperlink{Etingof}{Etingof}, \hyperlink{BenZviNadler09}{Ben-Zvi \& Nadler 09}).

\hypertarget{RelationToGeometricRepresentationTheory}{}\subsubsection*{{Relation to geometric representation theory}}\label{RelationToGeometricRepresentationTheory}

For the moment see at \emph{[[Harish Chandra transform]]}.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[coherent D-module]], [[holonomic D-module]]


\item [[arithmetic D-module]]


\item [[linear differential equation]]


\item [[D-geometry]]


\item [[D-scheme]]


\item [[D-algebra]]


\item [[crystalline cohomology]]


\item [[Weyl algebra]], [[regular differential operator]], [[local system]], [[differential bimodule]], [[Grothendieck connection]], [[crystal]], [[algebraic analysis]].


\item [[Hecke category]], [[Harish Chandra transform]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

A comprehensive account is in chapter 2 of

\begin{itemize}%
\item [[Alexander Beilinson]] and [[Vladimir Drinfeld]], chapter 2 of \emph{[[Chiral Algebras]]}


\item [[Armand Borel]] et al., \emph{Algebraic D-modules}, Perspectives in Mathematics, Academic Press, 1987 (\href{http://www.math.columbia.edu/~scautis/dmodules/boreletal.djvu}{djvu})


\item R. Hotta, K. Takeuchi, T. Tanisaki, \emph{D-modules, perverse sheaves, and representation theory}, Progress in Mathematics \textbf{236}, Birkh\"a{}user (\href{http://www.math.columbia.edu/~scautis/dmodules/hottaetal.pdf}{pdf})



\end{itemize}
Discussion in [[derived algebraic geometry]] is in

\begin{itemize}%
\item [[Dennis Gaitsgory]], [[Nick Rozenblyum]], \emph{Crystals and D-modules}, Pure and Applied Mathematics Quarterly Volume 10 (2014) Number 1 (\href{http://arxiv.org/abs/1111.2087}{arXiv:1111.2087}, \href{http://www.intlpress.com/site/pub/pages/journals/items/pamq/content/vols/0010/0001/a002/index.html}{publisher})

\end{itemize}
Lecture notes include

\begin{itemize}%
\item [[Jacob Lurie]], \emph{Notes on crystals and algebraic $\mathcal{D}$-modules} ()


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


\item [[Joseph Bernstein]], \emph{Algebraic theory of D-modules} ([[BernsteinDModule.pdf:file]], \href{http://www.math.uchicago.edu/~mitya/langlands/Bernstein/Bernstein-dmod.ps}{ps}, \href{http://www.math.uchicago.edu/~mitya/langlands/Bernstein/Bernstein-dmod.dvi}{dvi})


\item Peter Schneiders' \href{http://www.analg.ulg.ac.be/jps/rec/idm.pdf}{notes},


\item Dragan Milii`s \href{http://www.math.utah.edu/~milicic/Eprints/dmodules.pdf}{notes}, , \href{http://www.math.utah.edu/~milicic/Eprints/book.pdf}{Localization and representation theory of reductive Lie groups};


\item [[Victor Ginzburg]]`s 1998 Chicago notes \href{http://www.math.harvard.edu/~gaitsgde/grad_2009/Ginzburg.pdf}{pdf}; A.


\item Braverman-T. Chmutova, \emph{Lectures on algebraic D-modules}, \href{http://www.math.harvard.edu/~gaitsgde/grad_2009/Dmod_brav.pdf}{pdf}


\item R. Bezrukavnikov, MIT course notes, \href{http://www.math.harvard.edu/~gaitsgde/grad_2009/bezr_notes.pdf}{pdf}


\item Notes in Gaitsgory's seminar \href{http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov17-19%28Crystals%29.pdf}{pdf}


\item A. [[Beilinson|Beĭlinson]], J. Bernstein, \emph{A proof of Jantzen's conjectures}, I. M. Gelfand Seminar, 1--50, Adv. Soviet Math., 16, Part 1, Amer. Math. Soc. 1993, \href{http://www.math.harvard.edu/~gaitsgde/grad_2009/BB%20-%20Jantzen.pdf}{pdf}



\end{itemize}
See also

\begin{itemize}%
\item [[Morihiko Saito]], \emph{Induced D-modules and differential complexes}, Bull. Soc. Math. France 117 (1989), 361--387, \href{http://smf4.emath.fr/Publications/Bulletin/117/pdf/smf_bull_117_361-387.pdf}{pdf}


\item D. Gieseker, \emph{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.


\item J.-E. Bj\"o{}rk, \emph{Rings of differential operators}, North-Holland Math. Library 21. North-Holland Publ. 1979. xvii+374 pp.


\item [[Masaki Kashiwara|M. Kashiwara]], W.Schmid, \emph{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, Birkh\"a{}user, 1994, pp. 457--488


\item [[M. Kashiwara]], \emph{D-modules and representation theory of Lie groups}, Annales de l'institut Fourier, 43 no. 5 (1993), p. 1597-1618, \href{http://aif.cedram.org/item?id=AIF_1993__43_5_1597_0}{article}, \href{http://www.ams.org/mathscinet-getitem?mr=95b:22033}{MR95b:22033}


\item P. Maisonobe, C. Sabbah, \emph{D-modules coh\'e{}rents et holonomes}, Travaux en cours, Hermann, Paris 1993. (collection of lecture notes)


\item [[Donu Arapura]], \emph{Notes on D-modules and connection with Hodge theory}, \href{http://www.math.purdue.edu/~dvb/preprints/dmod.pdf}{pdf}


\item Nero Budur, \emph{On the V-filtration of D-modules}, \href{http://arxiv.org/abs/math/0409123}{math.AG/0409123}, in ``Geometric methods in algebra and number theory'' Proc. 2003 conf. Univ. of Miami, edited by F. Bogomolov, Yu. Tschinkel



\end{itemize}
Review of [[six operations yoga]] for D-modules is in

\begin{itemize}%
\item [[Pavel Etingof]], \emph{Formalism of six functors on all (coherent) D-modules} (\href{http://www-math.mit.edu/~etingof/dmodfactsheet.pdf}{pdf})


\item [[David Ben-Zvi]], [[David Nadler]], section 3 of \emph{The Character Theory of a Complex Group} (\href{http://arxiv.org/abs/0904.1247}{arXiv:0904.1247})



\end{itemize}
See also

\begin{itemize}%
\item A. Beilinson, I. N. Bernstein, \emph{A proof of Jantzen conjecture}, Adv. in Soviet Math. 16, Part 1 (1993), 1-50. MR95a:22022

\end{itemize}
\hypertarget{blog_discussion}{}\subsection*{{Blog discussion}}\label{blog_discussion}

\begin{itemize}%
\item Secret Blogging Seminar \href{http://sbseminar.wordpress.com/2007/07/07/musings-on-d-modules/}{Musings on D-modules}, \href{http://sbseminar.wordpress.com/2007/07/14/musings-on-d-modules-part-2/}{Musings on D-modules, part 2}


\item The Everything Seminar \href{http://cornellmath.wordpress.com/2007/09/06/d-module-basics-i/}{D-module Basics I}, \href{http://cornellmath.wordpress.com/2007/09/09/d-module-basics-ii/}{D-Module Basics II}.



\end{itemize}
[[!redirects D-modules]]

[[!redirects twisted D-module]] [[!redirects twisted D-modules]]



\end{document}