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{{Short description|Module over a sheaf of differential operators}}
{{DISPLAYTITLE:''D''-module}}
{{needs inline citations|date=March 2025}}
In [[mathematics]], a '''''D''-module''' is a [[module (mathematics)|module]] over a [[ring (mathematics)|ring]] ''D'' of [[differential operator]]s. The major interest of such ''D''-modules is as an approach to the theory of [[linear partial differential equation]]s. Since around 1970, ''D''-module theory has been built up, mainly as a response to the ideas of [[Mikio Sato]] on [[algebraic analysis]], and expanding on the work of Sato and [[Joseph Bernstein]] on the [[Bernstein–Sato polynomial]].

Early major results were the Kashiwara constructibility theorem and Kashiwara index theorem of [[Masaki Kashiwara]]. The methods of ''D''-module theory have always been drawn from [[sheaf theory]] and other techniques with inspiration from the work of [[Alexander Grothendieck]] in [[algebraic geometry]]. This approach is global in character, and differs from the [[functional analysis]] techniques traditionally used to study differential operators. The strongest results are obtained for [[over-determined system]]s ([[holonomic system]]s), and on the [[characteristic variety]] cut out by the [[symbol of a differential operator|symbols]], which in the good case is a [[Lagrangian submanifold]] of the [[cotangent bundle]] of maximal dimension ([[involutive system]]s). The techniques were taken up from the side of the Grothendieck school by [[Zoghman Mebkhout]], who obtained a general, [[derived category]] version of the [[Riemann–Hilbert correspondence]] in all dimensions.

==Modules over the Weyl algebra==
The first case of algebraic ''D''-modules are modules over the [[Weyl algebra]] ''A''<sub>''n''</sub>(''K'') over a [[field (mathematics)|field]] ''K'' of [[characteristic (algebra)|characteristic]] zero. It is the algebra consisting of polynomials in the following variables 
:''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>, ∂<sub>1</sub>, ..., ∂<sub>''n''</sub>.
where the variables ''x''<sub>''i''</sub> and ∂<sub>''j''</sub> separately commute with each other, and ''x''<sub>''i''</sub> and ∂<sub>''j''</sub> commute for ''i'' ≠ ''j'', but the [[commutator]] satisfies the relation
:[∂<sub>''i''</sub>, ''x''<sub>''i''</sub>] = ∂<sub>''i''</sub>''x''<sub>''i''</sub> &minus; x<sub>''i''</sub>''∂''<sub>''i''</sub> = 1.
For any polynomial ''f''(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>), this implies the relation
:[∂<sub>''i''</sub>, ''f''] = ∂''f'' / ∂''x''<sub>''i''</sub>,
thereby relating the Weyl algebra to differential equations.

An (algebraic) ''D''-module is, by definition, a [[left module]] over the ring ''A''<sub>''n''</sub>(''K''). Examples for ''D''-modules include the Weyl algebra itself (acting on itself by left multiplication), the (commutative) [[polynomial ring]] ''K''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>], where ''x''<sub>''i''</sub> acts by multiplication and ∂<sub>''j''</sub> acts by [[partial differentiation]] with respect to ''x''<sub>''j''</sub> and, in a similar vein, the ring <math>\mathcal O(\mathbf C^n)</math> of holomorphic functions on '''C'''<sup>''n''</sup> (functions of ''n'' complex variables.)

Given some [[differential operator]] {{nowrap begin}}''P'' = ''a''<sub>''n''</sub>(''x'') ∂<sup>''n''</sup> + ...  + ''a''<sub>1</sub>(''x'') ∂<sup>1</sup> + ''a''<sub>0</sub>(''x''),{{nowrap end}} where ''x'' is a complex variable, ''a''<sub>''i''</sub>(''x'') are polynomials, the quotient module ''M'' = ''A''<sub>1</sub>('''C''')/''A''<sub>1</sub>('''C''')''P'' is closely linked to space of solutions of the differential equation
:''P f'' = 0,
where ''f'' is some [[holomorphic function]] in '''C''', say. The [[vector space]] consisting of the solutions of that equation is given by the space of homomorphisms of ''D''-modules <math>\mathrm{Hom} (M, \mathcal O(\mathbf C))</math>.

==''D''-modules on algebraic varieties==
The general theory of ''D''-modules is developed on a [[smooth morphism|smooth]] [[algebraic variety]] ''X'' defined over an algebraically closed field ''K'' of characteristic zero, such as ''K'' = '''C'''. The [[sheaf (mathematics)|sheaf]] of differential operators ''D''<sub>''X''</sub> is defined to be the ''O''<sub>''X''</sub>-algebra generated by the [[vector field]]s on ''X'', interpreted as [[differential algebra|derivations]]. A (left) ''D''<sub>''X''</sub>-module ''M'' is an ''O''<sub>''X''</sub>-module with a left [[Group action (mathematics)|action]] of ''D''<sub>''X''</sub> on it. Giving such an action is equivalent to specifying a ''K''-linear map
:<math>\nabla: D_X \rightarrow \operatorname{End}_K(M), v \mapsto \nabla_v</math>
satisfying
:<math>\nabla_{f v}(m) = f \, \nabla_v (m)</math>
:<math>\nabla_v (f m) = v(f) m + f \, \nabla_v (m)</math> ([[Product rule|Leibniz rule]])
:<math>\nabla_{[v, w]}(m) = [\nabla_v, \nabla_w](m)</math>
Here ''f'' is a regular function on ''X'', ''v'' and ''w'' are vector fields, <math>m \in M</math>, and [&minus;, &minus;] denotes the [[commutator]]. Therefore, if ''M'' is in addition a locally free ''O''<sub>''X''</sub>-module, giving ''M'' a ''D''-module structure is nothing else than equipping the [[vector bundle]] associated to ''M'' with a flat (or integrable) [[connection (vector bundle)|connection]].{{Sfn|Hotta|Takeuchi|Tanisaki|2008|p=18}}

As the ring ''D''<sub>''X''</sub> is noncommutative, left and right ''D''-modules have to be distinguished. However, the two notions can be exchanged, since there is an [[equivalence of categories]] between both types of modules, given by mapping a left module ''M'' to the [[tensor product]] ''M'' ⊗ Ω<sub>''X''</sub>, where Ω<sub>''X''</sub> is the [[line bundle]] given by the highest [[exterior power]] of [[differential form|differential 1-forms]] on ''X''. This bundle has a natural ''right'' action determined by
:&omega; ⋅ ''v'' := &minus; Lie<sub>''v''</sub> (&omega;),
where ''v'' is a differential operator of order one, that is to say a vector field, ω a ''n''-form (''n'' = dim ''X''), and Lie denotes the [[Lie derivative]].{{Sfn|Hotta|Takeuchi|Tanisaki|2008|pages=18–21}}

Locally, after choosing some [[Regular system of parameters|system of coordinates]] ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub> (''n'' = dim ''X'') on ''X'', which determine a basis ∂<sub>1</sub>, ..., ∂<sub>''n''</sub> of the [[tangent space]] of ''X'', sections of ''D''<sub>''X''</sub> can be uniquely represented as expressions
:<math>\sum f_{i_1, \dots, i_n} \partial_1^{i_1} \cdots \partial_n^{i_n}</math>, where the <math>f_{i_1, \dots, i_n}</math> are [[regular function]]s on ''X''.
In particular, when ''X'' is the ''n''-dimensional [[affine space]], this ''D''<sub>''X''</sub> is the Weyl algebra in ''n'' variables.

Many basic properties of ''D''-modules are local and parallel the situation of [[coherent sheaf|coherent sheaves]]. This builds on the fact that ''D''<sub>''X''</sub> is a [[locally free sheaf]] of ''O''<sub>''X''</sub>-modules, albeit of infinite rank, as the above-mentioned ''O''<sub>''X''</sub>-basis shows. A ''D''<sub>''X''</sub>-module that is coherent as an ''O''<sub>''X''</sub>-module can be shown to be necessarily locally free (of finite rank).

===Functoriality===
''D''-modules on different algebraic varieties are connected by [[image functors for sheaves|pullback and pushforward functors]] comparable to the ones for coherent sheaves. For a [[morphism of schemes|map]] ''f'': ''X'' → ''Y'' of smooth varieties, the definitions are this:
:''D''<sub>''X''&rarr;''Y''</sub> := ''O''<sub>''X''</sub> &otimes;<sub>''f''<sup>&minus;1</sup>(''O''<sub>''Y''</sub>)</sub> ''f''<sup>&minus;1</sup>(''D''<sub>''Y''</sub>)
This is equipped with a left ''D''<sub>''X''</sub> action in a way that emulates the [[chain rule]], and with the natural right action of ''f''<sup>&minus;1</sup>(''D''<sub>''Y''</sub>). The pullback is defined as
:''f''<sup>&lowast;</sup>(''M'') := ''D''<sub>''X''&rarr;''Y''</sub> &otimes;<sub>''f''<sup>&minus;1</sup>(''D''<sub>''Y''</sub>)</sub> ''f''<sup>&minus;1</sup>(''M'').
Here ''M'' is a left ''D''<sub>''Y''</sub>-module, while its pullback is a left module over ''X''. This functor is [[right exact functor|right exact]], its left [[derived functor]] is denoted L''f''<sup>∗</sup>. Conversely, for a right ''D''<sub>''X''</sub>-module ''N'', 
:''f''<sub>&lowast;</sub>(''N'') := ''f''<sub>&lowast;</sub>(''N'' &otimes;<sub>''D''<sub>''X''</sub></sub>  ''D''<sub>''X''&rarr;''Y''</sub>)
is a right ''D''<sub>''Y''</sub>-module. Since this mixes the right exact tensor product with the left exact pushforward, it is common to set instead
:''f''<sub>&lowast;</sub>(''N'') := R''f''<sub>&lowast;</sub>(''N'' &otimes;<sup>L</sup><sub>''D''<sub>''X''</sub></sub>  ''D''<sub>''X''&rarr;''Y''</sub>).
Because of this, much of the theory of ''D''-modules is developed using the full power of [[homological algebra]], in particular [[derived category|derived categories]].


==Holonomic modules==

===Holonomic modules over the Weyl algebra===
It can be shown that the Weyl algebra is a (left and right) [[Noetherian ring]]. Moreover, it is [[simple ring|simple]], that is to say, its only two-sided [[ideal (algebra)|ideal]] are the [[zero ideal]] and the whole ring. These properties make the study of ''D''-modules manageable. Notably, standard notions from [[commutative algebra]] such as [[Hilbert polynomial]], multiplicity and [[length of a module|length]] of modules carry over to ''D''-modules. More precisely, ''D''<sub>''X''</sub> is equipped with the ''Bernstein filtration'', that is, the [[filtration (algebra)|filtration]] such that ''F''<sup>''p''</sup>''A''<sub>''n''</sub>(''K'') consists of ''K''-linear combinations of differential operators ''x''<sup>''α''</sup>∂<sup>''β''</sup> with |''α''|&nbsp;+&nbsp;|''β''|&nbsp;≤&nbsp;''p'' (using [[multiindex notation]]). The associated [[graded ring]] is seen to be isomorphic to the polynomial ring in 2''n'' indeterminates. In particular it is commutative.

[[finitely generated module|Finitely generated]] ''D''-modules ''M'' are endowed with so-called "good" filtrations ''F''<sup>∗</sup>''M'', which are ones compatible with ''F''<sup>∗</sup>''A''<sub>''n''</sub>(''K''), essentially parallel to the situation of the [[Artin–Rees lemma]]. The Hilbert polynomial is defined to be the [[numerical polynomial]] that agrees with the function
:''n'' ↦ dim<sub>''K''</sub> ''F''<sup>''n''</sup>''M''
for large ''n''. The dimension ''d''(''M'') of an  ''A''<sub>''n''</sub>(''K'')-module ''M'' is defined to be the degree of the Hilbert polynomial. It is bounded by the ''Bernstein inequality''
:''n'' &le; ''d''(''M'') &le; 2''n''.

A module whose dimension attains the least possible value, ''n'', is called ''holonomic''.

The ''A''<sub>1</sub>(''K'')-module ''M'' = ''A''<sub>1</sub>(''K'')/''A''<sub>1</sub>(''K'')''P'' (see above) is holonomic for any nonzero differential operator ''P'', but a similar claim for higher-dimensional Weyl algebras does not hold.

===General definition===
As mentioned above, modules over the Weyl algebra correspond to ''D''-modules on affine space. The Bernstein filtration not being available on ''D''<sub>''X''</sub> for general varieties ''X'', the definition is generalized to arbitrary affine smooth varieties ''X'' by means of ''order filtration'' on ''D''<sub>''X''</sub>, defined by the [[order of a differential operator|order of differential operators]]. The associated graded ring gr ''D''<sub>''X''</sub> is given by regular functions on the [[cotangent bundle]] T<sup>∗</sup>''X''.

The ''[[characteristic variety]]'' is defined to be the subvariety of the [[cotangent bundle]] cut out by the [[radical of an ideal|radical]] of the [[annihilator (ring theory)|annihilator]] of gr ''M'', where again ''M'' is equipped with a suitable filtration (with respect to the order filtration on ''D''<sub>''X''</sub>). As usual, the affine construction then glues to arbitrary varieties.

The Bernstein inequality continues to hold for any (smooth) variety ''X''. While the upper bound is an immediate consequence of the above interpretation of {{nowrap|gr ''D''<sub>''X''</sub>}} in terms of the cotangent bundle, the lower bound is more subtle.

===Properties and characterizations===
Holonomic modules have a tendency to behave like finite-dimensional vector spaces. For example, their length is finite. Also, ''M'' is holonomic if and only if all cohomology groups of the complex L''i''<sup>∗</sup>(''M'') are finite-dimensional ''K''-vector spaces, where ''i'' is the [[closed immersion]] of any point of ''X''.

For any ''D''-module ''M'', the ''dual module'' is defined by
:<math>\mathrm D(M) := \mathcal R \operatorname{Hom} (M, D_X) \otimes \Omega^{-1}_X [\dim X].</math>
Holonomic modules can also be characterized by a [[homological algebra|homological]] condition: ''M'' is holonomic if and only if D(''M'') is concentrated (seen as an object in the derived category of ''D''-modules) in degree 0. This fact is a first glimpse of [[Verdier duality]] and the [[Riemann–Hilbert correspondence]]. It is proven by extending the homological study of [[regular ring]]s (especially what is related to [[global homological dimension]]) to the filtered ring ''D''<sub>''X''</sub>.

Another characterization of holonomic modules is via [[symplectic geometry]]. The characteristic variety Ch(''M'') of any ''D''-module ''M'' is, seen as a subvariety of the cotangent bundle T<sup>∗</sup>''X'' of ''X'', an [[involutive system|involutive]] variety. The module is holonomic if and only if Ch(''M'') is [[Lagrangian submanifold|Lagrangian]].

==Applications==
One of the early applications of holonomic ''D''-modules was the [[Bernstein–Sato polynomial]].

===Kazhdan–Lusztig conjecture===
The [[Kazhdan–Lusztig conjecture]] was proved using ''D''-modules.

===Riemann–Hilbert correspondence===
The [[Riemann–Hilbert correspondence]] establishes a link between certain ''D''-modules and constructible sheaves. As such, it provided a motivation for introducing [[perverse sheaf|perverse sheaves]].

===Geometric representation theory===

''D''-modules are also applied in geometric representation theory. A main result in this area is the [[Beilinson–Bernstein localization]]. It relates ''D''-modules on [[flag variety|flag varieties]] ''G''/''B'' to representations of the [[Lie algebra]] <math>\mathfrak g</math> of a [[reductive group]] ''G''.
''D''-modules are also crucial in the formulation of the [[geometric Langlands program]].

==Notes==
{{reflist}}

==Bibliography==
* {{Citation | last1=Beilinson | first1=A. A. | author1-link=Alexander Beilinson | last2=Bernstein | first2=Joseph | author2-link=Joseph Bernstein | title=Localisation de ''g''-modules | mr=610137 | year=1981 | journal=Comptes Rendus de l'Académie des Sciences, Série I | issn=0249-6291 | volume=292 | issue=1 | pages=15–18}}
* {{Citation | last1=Björk | first1=J.-E. | title=Rings of differential operators | publisher=North-Holland | location=Amsterdam | series=North-Holland Mathematical Library | isbn=978-0-444-85292-2 | mr=549189 | year=1979 | volume=21}}
* {{Citation | last1=Brylinski | first1=Jean-Luc | last2=Kashiwara | first2=Masaki | author2-link=Masaki Kashiwara | title=Kazhdan–Lusztig conjecture and holonomic systems | doi=10.1007/BF01389272 | mr=632980 | year=1981 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=64 | issue=3 | pages=387–410| bibcode=1981InMat..64..387B | s2cid=18403883 }}
* {{Citation | last1=Coutinho | first1=S. C. | title=A primer of algebraic ''D''-modules | publisher=[[Cambridge University Press]] | series=London Mathematical Society Student Texts | isbn=978-0-521-55119-9 | mr=1356713 | year=1995 | volume=33}}
* {{Citation | editor1-last=Borel | editor1-first=Armand | editor1-link=Armand Borel | title=Algebraic D-Modules | publisher=[[Academic Press]] | location=Boston, MA | series=Perspectives in Mathematics | isbn=978-0-12-117740-9 | year=1987 | volume=2}}
*{{springer|id=D/d030020|title=D-module|author=M.G.M. van Doorn}}
* {{Citation | last1=Hotta | first1=Ryoshi | last2=Takeuchi | first2=Kiyoshi | last3=Tanisaki | first3=Toshiyuki | title=''D''-modules, perverse sheaves, and representation theory | url=http://www.math.harvard.edu/~gaitsgde/grad_2009/Hotta.pdf | publisher=Birkhäuser Boston | location=Boston, MA | series=Progress in Mathematics | isbn=978-0-8176-4363-8 | mr=2357361 | year=2008 | volume=236 | access-date=2009-12-10 | archive-url=https://web.archive.org/web/20160303172822/http://www.math.harvard.edu/~gaitsgde/grad_2009/Hotta.pdf | archive-date=2016-03-03 | url-status=dead }}

==External links==
* {{Citation | last1=Bernstein | first1=Joseph | author1-link=Joseph Bernstein | title=Algebraic theory of ''D''-modules | url=http://www.math.columbia.edu/~khovanov/resources/Bernstein-dmod.pdf }}
* {{Citation | last1=Gaitsgory | first1=Dennis | title=Lectures on Geometric Representation Theory | url=http://www.math.harvard.edu/~gaitsgde/267y/catO.pdf | access-date=2011-12-14 | archive-url=https://web.archive.org/web/20150326155114/http://www.math.harvard.edu/~gaitsgde/267y/catO.pdf | archive-date=2015-03-26 | url-status=dead }}
* {{Citation | last1=Milicic | first1=Dragan | title=Lectures on the Algebraic Theory of ''D''-Modules | url=http://www.math.utah.edu/~milicic/}}

[[Category:Algebraic analysis]]
[[Category:Partial differential equations]]
[[Category:Sheaf theory]]