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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a ''D''-module is a
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...
over a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
''D'' of
differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
s. The major interest of such ''D''-modules is as an approach to the theory of
linear partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
s. Since around 1970, ''D''-module theory has been built up, mainly as a response to the ideas of
Mikio Sato is a Japanese mathematician known for founding the fields of algebraic analysis, hyperfunctions, and holonomic quantum fields. He is a professor at the Research Institute for Mathematical Sciences in Kyoto. Education Sato studied at the Unive ...
on
algebraic analysis Algebraic analysis is an area of mathematics that deals with systems of linear partial differential equations by using sheaf theory and complex analysis to study properties and generalizations of functions such as hyperfunctions and microfunctio ...
, and expanding on the work of Sato and
Joseph Bernstein Joseph Bernstein (sometimes spelled I. N. Bernshtein; he, יוס(י)ף נאומוביץ ברנשטיין; russian: Иосиф Наумович Бернштейн; born 18 April 1945) is a Soviet-born Israeli mathematician working at Tel Aviv Univ ...
on the
Bernstein–Sato polynomial In mathematics, the Bernstein–Sato polynomial is a polynomial related to differential operators, introduced independently by and , . It is also known as the b-function, the b-polynomial, and the Bernstein polynomial, though it is not related ...
. Early major results were the
Kashiwara constructibility theorem file:Kashiwara City Office, Osaka pref01.JPG, 270px, Kashiwara City Hall is a city located in Osaka Prefecture, Japan. , the city had an estimated population of 67,698 in 32007 households and a population density of . The total area of the city ...
and
Kashiwara index theorem 270px, Kashiwara City Hall is a city located in Osaka Prefecture, Japan. , the city had an estimated population of 67,698 in 32007 households and a population density of . The total area of the city is . Geography Kashiwara is located about ...
of
Masaki Kashiwara is a Japanese mathematician. He was a student of Mikio Sato at the University of Tokyo. Kashiwara made leading contributions towards algebraic analysis, microlocal analysis, D-module, ''D''-module theory, Hodge theory, sheaf theory and represent ...
. The methods of ''D''-module theory have always been drawn from
sheaf theory In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
and other techniques with inspiration from the work of Alexander Grothendieck in
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
. The approach is global in character, and differs from the
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
techniques traditionally used to study differential operators. The strongest results are obtained for
over-determined system In mathematics, a system of equations is considered overdetermined if there are more equations than unknowns. An overdetermined system is almost always inconsistent (it has no solution) when constructed with random coefficients. However, an ov ...
s (
holonomic system In classical mechanics, holonomic constraints are relations between the position variables (and possibly time) that can be expressed in the following form: :f(u_1, u_2, u_3,\ldots, u_n, t) = 0 where \ are the ''n'' generalized coordinates that d ...
s), and on the
characteristic variety In mathematical analysis, the characteristic variety of a microdifferential operator ''P'' is an algebraic variety that is the zero set of the principal symbol of ''P'' in the cotangent bundle. It is invariant under a quantized contact transformatio ...
cut out by the
symbols A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different conc ...
, which in the good case is a
Lagrangian submanifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sy ...
of the
cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may ...
of maximal dimension (
involutive system In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations. In modern geometric term ...
s). The techniques were taken up from the side of the Grothendieck school by
Zoghman Mebkhout Zoghman Mebkhout (born 1949 ) (مبخوت زغمان) is a French-Algerian mathematician. He is known for his work in algebraic analysis, geometry and representation theory, more precisely on the theory of ''D''-modules. Career Mebkhout is c ...
, who obtained a general,
derived category In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proce ...
version of the
Riemann–Hilbert correspondence In mathematics, the term Riemann–Hilbert correspondence refers to the correspondence between regular singular flat connections on algebraic vector bundles and representations of the fundamental group, and more generally to one of several generaliz ...
in all dimensions.


Introduction: modules over the Weyl algebra

The first case of algebraic ''D''-modules are modules over the
Weyl algebra In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form : f_m(X) \partial_X^m + f_(X) \partial_X^ + \cdots + f_1(X) \partial_X + f_0(X). More prec ...
''A''''n''(''K'') over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
''K'' of characteristic zero. It is the algebra consisting of polynomials in the following variables :''x''1, ..., ''x''''n'', ∂1, ..., ∂''n''. where the variables ''x''''i'' and ∂''j'' separately commute with each other, and ''x''''i'' and ∂''j'' commute for ''i'' ≠ ''j'', but the
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, a ...
satisfies the relation : ''i'', ''x''''i''= ∂''i''''x''''i'' − x''i''''∂''''i'' = 1. For any polynomial ''f''(''x''1, ..., ''x''''n''), this implies the relation : ''i'', ''f''= ∂''f'' / ∂''x''''i'', thereby relating the Weyl algebra to differential equations. An (algebraic) ''D''-module is, by definition, a
left module In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mo ...
over the ring ''A''''n''(''K''). Examples for ''D''-modules include the Weyl algebra itself (acting on itself by left multiplication), the (commutative)
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) ...
''K'' 'x''1, ..., ''x''''n'' where ''x''''i'' acts by multiplication and ∂''j'' acts by
partial differentiation In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
with respect to ''x''''j'' and, in a similar vein, the ring \mathcal O(\mathbf C^n) of holomorphic functions on C''n'' (functions of ''n'' complex variables.) Given some
differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
''P'' = ''a''''n''(''x'') ∂''n'' + ... + ''a''1(''x'') ∂1 + ''a''0(''x''), where ''x'' is a complex variable, ''a''''i''(''x'') are polynomials, the quotient module ''M'' = ''A''1(C)/''A''1(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 \mathrm (M, \mathcal O(\mathbf C)).


''D''-modules on algebraic varieties

The general theory of ''D''-modules is developed on a
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebraic ...
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
''X'' defined over an algebraically closed field ''K'' of characteristic zero, such as ''K'' = C. The
sheaf Sheaf may refer to: * Sheaf (agriculture), a bundle of harvested cereal stems * Sheaf (mathematics), a mathematical tool * Sheaf toss, a Scottish sport * River Sheaf, a tributary of River Don in England * ''The Sheaf'', a student-run newspaper ser ...
of differential operators ''D''''X'' is defined to be the ''O''''X''-algebra generated by the vector fields on ''X'', interpreted as
derivations Derivation may refer to: Language * Morphological derivation, a word-formation process * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars Law * Derivative work, in copyright law * Derivation proceeding, a proc ...
. A (left) ''D''''X''-module ''M'' is an ''O''''X''-module with a left
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
of ''D''''X'' on it. Giving such an action is equivalent to specifying a ''K''-linear map :\nabla: D_X \rightarrow \operatorname_K(M), v \mapsto \nabla_v satisfying :\nabla_(m) = f \, \nabla_v (m) :\nabla_v (f m) = v(f) m + f \, \nabla_v (m) (
Leibniz rule Leibniz's rule (named after Gottfried Wilhelm Leibniz) may refer to one of the following: * Product rule in differential calculus * General Leibniz rule, a generalization of the product rule * Leibniz integral rule * The alternating series test, al ...
) :\nabla_(m) = nabla_v, \nabla_wm) Here ''f'' is a regular function on ''X'', ''v'' and ''w'' are vector fields, ''m'' a local section of ''M'', minus;, −denotes the
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, a ...
. Therefore, if ''M'' is in addition a locally free ''O''''X''-module, giving ''M'' a ''D''-module structure is nothing else than equipping the
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
associated to ''M'' with a flat (or integrable) connection. As the ring ''D''''X'' 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 In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences fro ...
between both types of modules, given by mapping a left module ''M'' to the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
''M'' ⊗ Ω''X'', where Ω''X'' is the
line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...
given by the highest
exterior power In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
of differential 1-forms on ''X''. This bundle has a natural ''right'' action determined by :ω ⋅ ''v'' := − Lie''v'' (ω), 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 In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...
. Locally, after choosing some
system of coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
''x''1, ..., ''x''''n'' (''n'' = dim ''X'') on ''X'', which determine a basis ∂1, ..., ∂''n'' of the
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
of ''X'', sections of ''D''''X'' can be uniquely represented as expressions :\sum f_ \partial_1^ \cdots \partial_n^, where the f_ are
regular function In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regular ...
s on ''X''. In particular, when ''X'' is the ''n''-dimensional
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
, this ''D''''X'' is the Weyl algebra in ''n'' variables. Many basic properties of ''D''-modules are local and parallel the situation of
coherent sheaves In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
. This builds on the fact that ''D''''X'' is a
locally free sheaf In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
of ''O''''X''-modules, albeit of infinite rank, as the above-mentioned ''O''''X''-basis shows. A ''D''''X''-module that is coherent as an ''O''''X''-module can be shown to be necessarily locally free (of finite rank).


Functoriality

''D''-modules on different algebraic varieties are connected by pullback and pushforward functors comparable to the ones for coherent sheaves. For a
map A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
''f'': ''X'' → ''Y'' of smooth varieties, the definitions are this: :''D''''X''→''Y'' := ''O''''X''''f''−1(''O''''Y'') ''f''−1(''D''''Y'') This is equipped with a left ''D''''X'' action in a way that emulates the
chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
, and with the natural right action of ''f''−1(''D''''Y''). The pullback is defined as :''f''(''M'') := ''D''''X''→''Y''''f''−1(''D''''Y'') ''f''−1(''M''). Here ''M'' is a left ''D''''Y''-module, while its pullback is a left module over ''X''. This functor is right exact, its left
derived functor In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. Motivation It was noted in vari ...
is denoted L''f''. Conversely, for a right ''D''''X''-module ''N'', :''f''(''N'') := ''f''(''N'' ⊗''D''''X'' ''D''''X''→''Y'') is a right ''D''''Y''-module. Since this mixes the right exact tensor product with the left exact pushforward, it is common to set instead :''f''(''N'') := R''f''(''N'' ⊗L''D''''X'' ''D''''X''→''Y''). Because of this, much of the theory of ''D''-modules is developed using the full power of
homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
, in particular
derived categories In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proce ...
.


Holonomic modules


Holonomic modules over the Weyl algebra

It can be shown that the Weyl algebra is a (left and right)
Noetherian ring In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noether ...
. Moreover, it is
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
, that is to say, its only two-sided
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
are the
zero ideal In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context. Additive identities An additive identi ...
and the whole ring. These properties make the study of ''D''-modules manageable. Notably, standard notions from
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...
such as
Hilbert polynomial In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which measure the growth of the dimension of the homoge ...
, multiplicity and
length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...
of modules carry over to ''D''-modules. More precisely, ''D''''X'' is equipped with the ''Bernstein filtration'', that is, the
filtration Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter ...
such that ''F''''p''''A''''n''(''K'') consists of ''K''-linear combinations of differential operators ''x''''α''''β'' with , ''α'',  + , ''β'',  ≤ ''p'' (using multiindex notation). The associated
graded ring In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the se ...
is seen to be isomorphic to the polynomial ring in 2''n'' indeterminates. In particular it is commutative. Finitely generated ''D''-modules ''M'' are endowed with so-called "good" filtrations ''F''''M'', which are ones compatible with ''F''''A''''n''(''K''), essentially parallel to the situation of the
Artin–Rees lemma In mathematics, the Artin–Rees lemma is a basic result about modules over a Noetherian ring, along with results such as the Hilbert basis theorem. It was proved in the 1950s in independent works by the mathematicians Emil Artin and David Re ...
. The Hilbert polynomial is defined to be the
numerical polynomial In mathematics, an integer-valued polynomial (also known as a numerical polynomial) P(t) is a polynomial whose value P(n) is an integer for every integer ''n''. Every polynomial with integer coefficients is integer-valued, but the converse is not tr ...
that agrees with the function :''n'' ↦ dim''K'' ''F''''n''''M'' for large ''n''. The dimension ''d''(''M'') of an ''A''''n''(''K'')-module ''M'' is defined to be the degree of the Hilbert polynomial. It is bounded by the ''Bernstein inequality'' :''n'' ≤ ''d''(''M'') ≤ 2''n''. A module whose dimension attains the least possible value, ''n'', is called ''holonomic''. The ''A''1(''K'')-module ''M'' = ''A''1(''K'')/''A''1(''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''''X'' for general varieties ''X'', the definition is generalized to arbitrary affine smooth varieties ''X'' by means of ''order filtration'' on ''D''''X'', defined by the order of differential operators. The associated graded ring gr ''D''''X'' is given by regular functions on the
cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may ...
T''X''. The ''
characteristic variety In mathematical analysis, the characteristic variety of a microdifferential operator ''P'' is an algebraic variety that is the zero set of the principal symbol of ''P'' in the cotangent bundle. It is invariant under a quantized contact transformatio ...
'' is defined to be the subvariety of the
cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may ...
cut out by the
radical Radical may refer to: Politics and ideology Politics *Radical politics, the political intent of fundamental societal change *Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe and ...
of the annihilator of gr ''M'', where again ''M'' is equipped with a suitable filtration (with respect to the order filtration on ''D''''X''). 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 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''(''M'') are finite-dimensional ''K''-vector spaces, where ''i'' is the
closed immersion In algebraic geometry, a closed immersion of schemes is a morphism of schemes f: Z \to X that identifies ''Z'' as a closed subset of ''X'' such that locally, regular functions on ''Z'' can be extended to ''X''. The latter condition can be formaliz ...
of any point of ''X''. For any ''D''-module ''M'', the ''dual module'' is defined by :\mathrm D(M) := \mathcal R \operatorname (M, D_X) \otimes \Omega^_X dim X Holonomic modules can also be characterized by a
homological Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor *Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chromo ...
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 In mathematics, Verdier duality is a cohomology, cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by as an analog for locally compact space, locally compact topolo ...
and the
Riemann–Hilbert correspondence In mathematics, the term Riemann–Hilbert correspondence refers to the correspondence between regular singular flat connections on algebraic vector bundles and representations of the fundamental group, and more generally to one of several generaliz ...
. It is proven by extending the homological study of
regular ring In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let ''A'' be a Noetherian local ring with maximal ide ...
s (especially what is related to
global homological dimension In ring theory and homological algebra, the global dimension (or global homological dimension; sometimes just called homological dimension) of a ring (mathematics), ring ''A'' denoted gl dim ''A'', is a non-negative integer or infinity which is a ho ...
) to the filtered ring ''D''''X''. Another characterization of holonomic modules is via
symplectic geometry Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed differential form, closed, nondegenerate form, nondegenerate different ...
. The characteristic variety Ch(''M'') of any ''D''-module ''M'' is, seen as a subvariety of the cotangent bundle T''X'' of ''X'', an involutive variety. The module is holonomic if and only if Ch(''M'') is
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
.


Applications

One of the early applications of holonomic ''D''-modules was the
Bernstein–Sato polynomial In mathematics, the Bernstein–Sato polynomial is a polynomial related to differential operators, introduced independently by and , . It is also known as the b-function, the b-polynomial, and the Bernstein polynomial, though it is not related ...
.


Kazhdan–Lusztig conjecture

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


Riemann–Hilbert correspondence

The
Riemann–Hilbert correspondence In mathematics, the term Riemann–Hilbert correspondence refers to the correspondence between regular singular flat connections on algebraic vector bundles and representations of the fundamental group, and more generally to one of several generaliz ...
establishes a link between certain ''D''-modules and constructible sheaves. As such, it provided a motivation for introducing
perverse sheaves The mathematical term perverse sheaves refers to a certain abelian category associated to a topological space ''X'', which may be a real or complex manifold, or a more general topologically stratified space, usually singular. This concept was intro ...
.


Geometric representation theory

''D''-modules are also applied in
geometric representation theory Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
. A main result in this area is the
Beilinson–Bernstein localization In mathematics, especially in representation theory and algebraic geometry, the Beilinson–Bernstein localization theorem relates D-modules on flag variety, flag varieties ''G''/''B'' to representations of the Lie algebra \mathfrak g attached to a ...
. It relates ''D''-modules on
flag varieties In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flag (linear algebra), flags in a finite-dimensional vector space ''V'' over a field (mathematics), field F. When F is the real or complex nu ...
''G''/''B'' to representations of the
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
\mathfrak g of a
reductive group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...
''G''. ''D''-modules are also crucial in the formulation of the
geometric Langlands program In mathematics, the geometric Langlands correspondence is a reformulation of the Langlands correspondence obtained by replacing the number fields appearing in the original number theory, number theoretic version by function field of an algebraic var ...
.


References

* * * * * * *


External links

* * * {{Citation , last1=Milicic , first1=Dragan , title=Lectures on the Algebraic Theory of ''D''-Modules , url=http://www.math.utah.edu/~milicic/ Algebraic analysis Partial differential equations Sheaf theory