|
In mathematics, a D-module is a module over a ring ''D'' of differential operators. The major interest of such D-modules is as an approach to the theory of linear partial differential equations. 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. 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 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. ==Introduction: modules over the Weyl algebra== The first case of algebraic ''D''-modules 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 the variables ''x''''i'' and ∂''j'' separately commute with each other, and ''x''''i'' and ∂''j'' commute for ''i'' ≠ ''j'', but the commutator satisfies the relation :(''x''''i'' ) = ∂''i''''x''''i'' − x''i''''∂''''i'' = 1. For any polynomial ''f''(''x''1, ..., ''x''''n''), this implies the relation :(''f'' ) = ∂''f'' / ∂''x''''i'', thereby relating the Weyl algebra to differential equations. An (algebraic) ''D''-module is, by definition, a left module 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 ''K''(..., ''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'' (functions of ''n'' complex variables.) 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 ''D''-modules . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「D-module」の詳細全文を読む スポンサード リンク
|