Dmodule

In mathematics, a Dmodule is a module over a ring D of differential operators. The major interest of such Dmodules is as an approach to the theory of linear partial differential equations. Since around 1970, Dmodule 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 Dmodule 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 overdetermined systems (holonomic systems), and on the characteristic variety cut out by the symbols, in the good case for which it is a Lagrangian submanifold of the cotangent bundle of maximal dimension (involutive systems). 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.
Contents
Introduction: modules over the Weyl algebra
The first case of algebraic Dmodules are modules over the Weyl algebra A_{n}(K) over a field K of characteristic zero. It is the algebra consisting of polynomials in the following variables
 x_{1}, ..., x_{n}, ∂_{1}, ..., ∂_{n}.
where all of the variables x_{i} and ∂_{j} commute with each other, but the commutator
 [∂_{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) Dmodule is, by definition, a left module over the ring A_{n}(K). Examples for Dmodules include the Weyl algebra itself (acting on itself by left multiplication), the (commutative) polynomial ring K[x_{1}, ..., x_{n}], where x_{i} acts by multiplication and ∂_{j} acts by partial differentiation with respect to x_{j} and, in a similar vein, the ring of holomorphic functions on C^{n}, the complex plane.
Given some differential operator 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 Dmodules .
Dmodules on algebraic varieties
The general theory of Dmodules is developed on a smooth algebraic variety X defined over an algebraically closed field K of characteristic zero, such as K = C. The sheaf of differential operators D_{X} is defined to be the O_{X}module generated by the vector fields on X, interpreted as derivations. A (left) D_{X}module M is an O_{X}module with a left action of D_{X}. Giving such an action is equivalent to specifying a Klinear map
satisfying
 (Leibniz rule)
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_{X}module, giving M a Dmodule structure is nothing else than equipping the vector bundle associated to M with a flat (or integrable) connection.
As the ring D_{X} is noncommutative, left and right Dmodules 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_{X}, where Ω_{X} is the line bundle given by the highest exterior power of differential 1forms 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 nform (n = dim X), and Lie denotes the Lie derivative.
Locally, after choosing some system of coordinates x_{1}, ..., x_{n} (n = dim X) on X, which determine a basis ∂_{1}, ..., ∂_{n} of the tangent space of X, sections of D_{X} can be uniquely represented as expressions
 , where the are regular functions on X.
In particular, when X is the ndimensional affine space, this D_{X} is the Weyl algebra in n variables.
Many basic properties of Dmodules are local and parallel the situation of coherent sheaves. This builds on the fact that D_{X} is a locally free sheaf of O_{X}modules, albeit of infinite rank, as the abovementioned 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
Dmodules on different algebraic varieties are connected by pullback and pushforward functors comparable to the ones for coherent sheaves. For a map f: X → Y of smooth varieties, the definitions are this:
 D_{X→Y} := O_{X} ⊗_{f−1(OX)} f^{−1}(D_{X})
This is equipped with a left D_{X} action in a way that emulates the chain rule, and with the natural right action of f^{−1}(D_{X}). The pullback is defined as
 f^{∗}(M) := D_{X→Y} ⊗_{f−1(DX)} 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 is denoted Lf^{∗}. Conversely, for a right D_{X}module N,
 f_{∗}(N) := f_{∗}(N ⊗_{DX} 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) := Rf_{∗}(N ⊗^{L}_{DX} D_{X→Y}).
Because of this, much of the theory of Dmodules is developed using the full power of homological algebra, in particular 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, that is to say, its only left and right ideal are the zero ideal and the whole ring. These properties make the study of Dmodules manageable. Notably, standard notions from commutative algebra such as Hilbert polynomial, multiplicity and length of modules carry over to Dmodules. More precisely, D_{X} is equipped with the Bernstein filtration, that is, the filtration such that F^{p}A_{n}(K) consists of Klinear combinations of differential operators x^{α}∂^{β} with α+β ≤ p (using multiindex notation). The associated graded ring is seen to be isomorphic to the polynomial ring in 2n indeterminates. In particular it is commutative.
Finitely generated Dmodules M are endowed with socalled "good" filtrations F^{∗}M, which are ones compatible with F^{∗}A_{n}(K), essentially parallel to the situation of the ArtinRees lemma. The Hilbert polynomial is defined to be the numerical polynomial that agrees with the function
 n ↦ dim_{K} F^{n}M
for large n. The dimension d(M) of a 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) ≤ 2n.
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 higherdimensional Weyl algebras does not hold.
General definition
As mentioned above, modules over the Weyl algebra correspond to Dmodules 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 T^{∗}X.
The characteristic variety is defined to be the subvariety of the cotangent bundle cut out by the radical 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 gr D_{X} in terms of the cotangent bundle, the lower bound is more subtle.
Properties and characterizations
Holonomic modules have a tendency to behave like finitedimensional vector spaces. For example, their length is finite. Also, M is holonomic if and only if all cohomology groups of the complex Li^{∗}(M) are finitedimensional Kvector spaces, where i is the closed immersion of any point of X.
For any Dmodule M, the dual module is defined by
Holonomic modules can also be characterized by a homological condition: M is holonomic if and only if D(M) is concentrated (seen as an object in the derived category of Dmodules) 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 rings (especially what is related to global homological dimension) to the filtered ring D_{X}.
Another characterization of holonomic modules is via symplectic geometry. The characteristic variety Ch(M) of any Dmodule 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.
Applications
One of the early applications of holonomic Dmodules was the Bernstein–Sato polynomial.
Kazhdan–Lusztig conjecture
The Kazhdan–Lusztig conjecture was proved using Dmodules.
Riemann–Hilbert correspondence
The Riemann–Hilbert correspondence establishes a link between certain Dmodules and constructible sheaves. As such, it provided a motivation for introducing perverse sheaves.
References
 Beilinson, A. A.; Bernstein, Joseph (1981), "Localisation de gmodules", Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique 292 (1): 15–18, ISSN 02496291, MR610137
 Björk, J.E. (1979), Rings of differential operators, NorthHolland Mathematical Library, 21, Amsterdam: NorthHolland, ISBN 9780444852922, MR549189
 Brylinski, JeanLuc; Kashiwara, Masaki (1981), "Kazhdan–Lusztig conjecture and holonomic systems", Inventiones Mathematicae 64 (3): 387–410, doi:10.1007/BF01389272, ISSN 00209910, MR632980
 Coutinho, S. C. (1995), A primer of algebraic Dmodules, London Mathematical Society Student Texts, 33, Cambridge University Press, ISBN 9780521551199; 9780521559089, MR1356713
 Borel, Armand, ed. (1987), Algebraic DModules, Perspectives in Mathematics, 2, Boston, MA: Academic Press, ISBN 9780121177409
 M.G.M. van Doorn (2001), "Dmodule", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 9781556080104, http://eom.springer.de/D/d030020.htm
 Hotta, Ryoshi; Takeuchi, Kiyoshi; Tanisaki, Toshiyuki (2008), Dmodules, perverse sheaves, and representation theory, Progress in Mathematics, 236, Boston, MA: Birkhäuser Boston, ISBN 9780817643638, MR2357361, http://www.math.harvard.edu/~gaitsgde/grad_2009/Hotta.pdf
External links
Categories: Algebraic analysis
 Partial differential equations
 Sheaf theory
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