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| 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]].
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| 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]]. The 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]], in the good case for which it 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.
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| ==Introduction: modules over the Weyl algebra==
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| 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
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| :''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>, ∂<sub>1</sub>, ..., ∂<sub>''n''</sub>.
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| where all of the variables ''x''<sub>''i''</sub> and ∂<sub>''j''</sub> commute with each other, but the [[commutator]]
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| :[∂<sub>''i''</sub>, ''x''<sub>''i''</sub>] = ∂<sub>''i''</sub>''x''<sub>''i''</sub> − x<sub>''i''</sub>''∂''<sub>''i''</sub> = 1.
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| For any polynomial ''f''(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>), this implies the relation
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| :[∂<sub>''i''</sub>, ''f''] = ∂''f'' / ∂''x''<sub>''i''</sub>,
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| thereby relating the Weyl algebra to differential equations.
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| 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>, the complex plane.
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| 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
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| :''P f'' = 0,
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| 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>.
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| ==''D''-modules on algebraic varieties==
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| 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|action]] of ''D''<sub>''X''</sub>. Giving such an action is equivalent to specifying a ''K''-linear map
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| :<math>\nabla: D_X \rightarrow End_K(M), v \mapsto \nabla_v</math>
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| satisfying
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| :<math>\nabla_{f v}(m) = f \nabla_v (m)</math>
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| :<math>\nabla_v (f m) = v(f) m + f \nabla_v (m)</math> ([[Leibniz rule]])
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| :<math>\nabla_{[v, w]}(m) = [\nabla_{v}, \nabla_{w}](m)</math>
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| Here ''f'' is a regular function on ''X'', ''v'' and ''w'' are vector fields, ''m'' a local section of ''M'', [−, −] 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]].
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| 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
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| :ω ⋅ ''v'' := − Lie<sub>''v''</sub> (ω),
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| 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]].
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| 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
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| :<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''.
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| In particular, when ''X'' is the ''n''-dimensional [[affine space]], this ''D''<sub>''X''</sub> is the Weyl algebra in ''n'' variables.
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| 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).
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| ===Functoriality===
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| ''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:
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| :''D''<sub>''X''→''Y''</sub> := ''O''<sub>''X''</sub> ⊗<sub>''f''<sup>−1</sup>(''O''<sub>''Y''</sub>)</sub> ''f''<sup>−1</sup>(''D''<sub>''Y''</sub>)
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| 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>−1</sup>(''D''<sub>''Y''</sub>). The pullback is defined as
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| :''f''<sup>∗</sup>(''M'') := ''D''<sub>''X''→''Y''</sub> ⊗<sub>''f''<sup>−1</sup>(''D''<sub>''Y''</sub>)</sub> ''f''<sup>−1</sup>(''M'').
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| 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'',
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| :''f''<sub>∗</sub>(''N'') := ''f''<sub>∗</sub>(''N'' ⊗<sub>''D''<sub>''X''</sub></sub> ''D''<sub>''X''→''Y''</sub>)
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| 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
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| :''f''<sub>∗</sub>(''N'') := R''f''<sub>∗</sub>(''N'' ⊗<sup>L</sup><sub>''D''<sub>''X''</sub></sub> ''D''<sub>''X''→''Y''</sub>).
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| 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]].
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| ==Holonomic modules==
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| ===Holonomic modules over the Weyl algebra===
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| 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|+|β| ≤ ''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.
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| [[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
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| :''n'' ↦ dim<sub>''K''</sub> ''F''<sup>''n''</sup>''M''
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| for large ''n''. The dimension ''d''(''M'') of a ''A''<sub>''n''</sub>(''K'')-module ''M'' is defined to be the degree of the Hilbert polynomial. It is bounded by the ''Bernstein inequality''
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| :''n'' ≤ ''d''(''M'') ≤ 2''n''.
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| A module whose dimension attains the least possible value, ''n'', is called ''holonomic''.
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| 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.
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| ===General definition===
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| 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''.
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| 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.
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| 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.
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| ===Properties and characterizations===
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| 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''.
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| For any ''D''-module ''M'', the ''dual module'' is defined by
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| :<math>\mathrm D(M) := \mathcal R \mathrm{Hom} (M, D_X) \otimes \Omega^{-1}_X [\operatorname{ dim} X].</math>
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| 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>.
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| 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]].
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| ==Applications==
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| One of the early applications of holonomic ''D''-modules was the [[Bernstein–Sato polynomial]].
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| ===Kazhdan–Lusztig conjecture===
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| The [[Kazhdan–Lusztig conjecture]] was proved using ''D''-modules.
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| ===Riemann–Hilbert correspondence===
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| 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]].
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| ==References==
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| * {{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 des Séances de l'Académie des Sciences. Série I. Mathématique | issn=0249-6291 | volume=292 | issue=1 | pages=15–18}}
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| * {{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}}
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| * {{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}}
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| * {{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}}
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| * {{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}}
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| *{{springer|id=D/d030020|title=D-module|author=M.G.M. van Doorn}}
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| * {{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}}
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| ==External links==
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| * {{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 }}
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| * {{Citation | last1=Gaitsgory | first1=Dennis | title=Lectures on Geometric Representation Theory | url=http://www.math.harvard.edu/~gaitsgde/267y/catO.pdf}}
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| * {{Citation | last1=Milicic | first1=Dragan | title=Lectures on the Algebraic Theory of ''D''-Modules | url=http://www.math.utah.edu/~milicic/}}
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| [[Category:Algebraic analysis]]
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| [[Category:Partial differential equations]]
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| [[Category:Sheaf theory]]
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