×

zbMATH — the first resource for mathematics

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).

MSC:
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
Software:
Risa/Asir; Kan
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Assi, A.; Castro-Jiménez, F.J.; Granger, J.M., How to calculate the slopes of a \(D\)-module, Compositio math., 104, 107-123, (1996) · Zbl 0862.32005
[2] Becker, T.; Weispfenning, V., Gröbner bass, (1993), Springer New York
[3] J. Bernstein, Algebraic theory of D-modules, unpublished notes.
[4] Björk, J.E., Rings of differential operators, (1979), North-Holland Amsterdam
[5] Borel, A., Algebraic D-modules, (1987), Academic Press Boston · Zbl 0642.32001
[6] Buchberger, B., Ein algorithmisches kriterium für die Lösbarkeit eines algebraischen gleichungssystems, Aequationes math., 4, 374-383, (1970) · Zbl 0212.06401
[7] F. Castro, Calculs Effectifs pour les Idéaux d’Opérateurs Différentiels, Travaux en Cours, vol. 24, Hermann, Paris, 1987, pp. 1-19.
[8] F.J. Castro-Jimenez, L. Narváez Macarro, Homogenizing differential operators, Prepublication No.36, Facultad de Matemáticas, Universidad de Sevilla, 1997.
[9] Cox, D.; Little, J.; O’Shea, D., Ideals, varieties and algorithms, (1991), Springer New York
[10] Eisenbud, D., Commutative algebra with a view toward algebraic geometry, (1995), Springer New York · Zbl 0819.13001
[11] Eisenbud, D.; Huneke, C.; Vasconcelos, W., Direct methods for primary decomposition, Invent. math., 110, 207-235, (1992) · Zbl 0770.13018
[12] A. Galligo, Some algorithmic questions on ideals of differential operators, Lecture Notes in Computer Science, vol. 204, Springer, Berlin, 1985, pp. 413-421. · Zbl 0634.16001
[13] R. Hartshorne, Residues and Duality, Lecture Notes in Mathematics, vol. 20, Springer, Berlin, 1966.
[14] Kashiwara, M., B-functions and holonomic systems – rationality of roots of b-functions, Invent. math., 38, 33-53, (1976) · Zbl 0354.35082
[15] Kashiwara, M., On the holonomic systems of linear differential equations, II, Invent. math., 49, 121-135, (1978) · Zbl 0401.32005
[16] M. Kashiwara, Vanishing cycle sheaves and holonomic systems of differential equations, Lecture Notes in Mathematics, vol. 1016, Springer, Berlin, 1983, pp. 134-142. · Zbl 0566.32022
[17] Kashiwara, M., Systems of microdifferential equations, (1983), Birkhäuser Boston · Zbl 0566.32022
[18] Kashiwara, M.; Kawai, T., On the characteristic variety of a holonomic system with regular singularities, Adv. math., 34, 163-184, (1979) · Zbl 0449.58019
[19] M. Kashiwara, T. Kawai, Second microlocalization and asymptotic expansions, Lecture Notes in Physics, vol. 126, Springer, Berlin, 1980, pp. 21-76. · Zbl 0458.46027
[20] Kashiwara, M.; Kawai, T., On holonomic systems of microdifferential equations, III, Publ. RIMS, Kyoto univ., 17, 813-979, (1981) · Zbl 0505.58033
[21] La Scala, R.; Stillman, M., Strategies for computing minimal free resolutions, J. symbolic comput., 26, 409-431, (1998) · Zbl 1034.68716
[22] Laurent, Y., Polygône de Newton et b-fonctions pour LES modules microdifferentiels, Ann. sci. éc. norm. sup., 20, 391-441, (1987) · Zbl 0646.58021
[23] Laurent, Y.; Monteiro Fernandes, T., Systèmes différentiels fuchsiens le long d’une sous-variété, Publ. RIMS, Kyoto univ., 24, 397-431, (1988) · Zbl 0704.35032
[24] Laurent, Y.; Schapira, P., Images inverses des modules différentiels, Compositio math., 61, 229-251, (1987) · Zbl 0617.32014
[25] Z. Mebkhout, Le Formalisme des Six Opérations de Grothendieck pour les \(DX\)-modules Cohérents, Travaux en Cours, vol. 35, Hermann, Paris, 1989.
[26] T. Mora, Seven variations on standard bases, preprint, Univ. Genova, 1988.
[27] M. Noro et al., Risa/Asir — a computer algebra system. Binary available at \(/\)\(/\)\(/\) (1993, 1995).
[28] Oaku, T., Computation of the characteristic variety and the singular locus of a system of differential equations with polynomial coefficients, Japan J. ind. appl. math., 11, 485-497, (1994) · Zbl 0811.35006
[29] Oaku, T., Algorithmic methods for Fuchsian systems of linear partial differential equations, J. math. soc. Japan, 47, 297-328, (1995) · Zbl 0847.35032
[30] Oaku, T., An algorithm of computing b-functions, Duke math. J., 87, 115-132, (1997) · Zbl 0893.32009
[31] Oaku, T., Algorithms for the b-function and D-modules associated with a polynomial, J. pure appl. algebra, 117 & 118, 495-518, (1997) · Zbl 0918.32006
[32] Oaku, T., Algorithms for b-functions, restrictions, and algebraic local cohomology groups of D-modules, Adv. in appl. math., 19, 61-105, (1997) · Zbl 0938.32005
[33] Oaku, T.; Takayama, N., An algorithm for de Rham cohomology groups of the complement of an affine variety, J. pure appl. algebra, 139, 201-233, (1999) · Zbl 0960.14008
[34] T. Oaku, N. Takayama, U. Walther, A localization algorithm for D-modules, J. Symbolic Comput., to appear. · Zbl 1012.13010
[35] Robbiano, L., On the theory of graded structures, J. symbolic comput., 2, 139-170, (1986) · Zbl 0609.13007
[36] M. Saito, B. Sturmfels, N. Takayama, Gröbner Deformations of Hypergeometric Differential Equations, Springer Verlag, Berlin, 2000. · Zbl 0946.13021
[37] Schapira, P., Microdifferential systems in the complex domain, (1985), Springer Berlin · Zbl 0554.32022
[38] Shimoyama, T.; Yokoyama, K., Localization and primary decomposition of polynomial ideals, J. symbolic comput., 22, 247-277, (1996) · Zbl 0874.13022
[39] Takayama, N., Gröbner basis and the problem of contiguous relations, Japan J. appl. math., 6, 147-160, (1989) · Zbl 0691.68032
[40] Takayama, N., An algorithm of constructing the integral of a module — an infinite dimensional analog of Gröbner basis, (), 206-211
[41] N. Takayama, Kan: A system for computation in algebraic analysis, Source code available at \(/\)\(/\)\(/\). Version 1 (1991), Version 2 (1994), the latest version is 2.981217 (1998).
[42] Walther, U., Algorithmic computation of local cohomology modules and the cohomological dimension of algebraic varieties, J. pure appl. algebra, 139, 303-321, (1999) · Zbl 0960.14003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.