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Algorithms for \(D\)-modules – restriction, tensor product, localization, and local cohomology groups. (English) Zbl 0983.13008
An algorithm is given, valid for a class of affine \(D\)-modules which includes the holonomic \(D\)-modules, for computing cohomology groups of the restriction of a \(D\)-module to a linear subvariety. This algorithm uses a free resolution of the \(D\)-module, whose construction in turn depends on the previously established Gröbner basis theory for the Weyl algebra . Two versions of constructing the free resolution are given, one of which has been implemented in the second author’s system kan since 1994, but is written up here for the first time. (The source code of kan is available from www.math.kobe-u.ac.jp/KAN.) As applications of the basic algorithm, algorithms for computing tensor product, localization, and algebraic local cohomology groups of holonomic systems are obtained. Such algorithms have been previously known only for very special cases.
A further interesting application is to calculation of certain de Rham cohomology groups; see the authors’ paper: T. Oaku and N. Takayama, J. Pure Appl. Algebra 139, 201-233 (1999; Zbl 0960.14008).

13N10 Commutative rings of differential operators and their modules
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
14Q20 Effectivity, complexity and computational aspects of algebraic geometry
32C38 Sheaves of differential operators and their modules, \(D\)-modules
14B15 Local cohomology and algebraic geometry
68W30 Symbolic computation and algebraic computation
Risa/Asir; Kan
Full Text: DOI arXiv
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