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Algebraic study of systems of partial differential equations. (Master’s thesis, Tokyo University, December 1970. Translated by Andrea D’Agnolo and Pierre Schneiders. With a foreword by Pierre Schapira). (English) Zbl 0877.35003
Mém. Soc. Math. Fr., Nouv. Sér. 63, 72 p. (1995); erratum ibid. 125, 313 (1997).
Summary: This Mémoire is a translation of M. Kashiwara’s thesis. In this pioneering work, the author initiates the study of systems of linear partial differential equations with analytic coefficients from the point of view of modules over the ring \(\mathcal D\) of differential operators. Following some preliminaries on good filtrations and non-commutative localization, the author introduces the notion of characteristic variety and of multiplicity of a \(\mathcal D\)-module. Then he shows that the classical Cauchy-Kovalevskaya theorem may be generalized as a formula for the solutions of non-characteristic inverse images of \(\mathcal D\)-modules. Among the applications of this result, we find a solvability criterion in the complex domain and a study of the Cauchy problem for hyperfunctions. The author also investigates the homological properties of \(\mathcal D\)-modules linking, in particular, their homological dimension to the codimension of their characteristic variety. The thesis concludes with an index formula for holonomic systems on smooth complex curves.

MSC:
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
58J15 Relations of PDEs on manifolds with hyperfunctions
32C38 Sheaves of differential operators and their modules, \(D\)-modules
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References:
[1] N. Bourbaki , Eléments de mathématique. Fascicule XXVIII. Algèbre commutative , Chapitre III, IV, Hermann, 1961 . · Zbl 0119.03603
[2] E. Cartan , Les systèmes différentiels extérieurs et leurs applications géométriques , Hermann, 1945 . MR 7,520d | Zbl 0063.00734 · Zbl 0063.00734
[3] H. Cartan and S. Eilenberg , Homological Algebra , Princeton University Press, 1956 . MR 17,1040e | Zbl 0075.24305 · Zbl 0075.24305
[4] J. Dieudonné and A. Grothendieck , Eléments de géométrie algébrique , Chapitre II, III, IV, Inst. Hautes Etudes Sci. Publ. Math. 8, 11, 17, 20, 24, 32. Numdam | Zbl 0136.15901 · Zbl 0136.15901
[5] J. Frisch , Points de platitude d’un morphisme d’espaces analytiques complexes , Invent. Math. 4 ( 1967 ), 118-138. MR 36 #5388 | Zbl 0167.06803 · Zbl 0167.06803
[6] R. Hartshorne , Residues and duality , Lecture Notes in Mathematics 20, Springer, 1966 . MR 36 #5145 | Zbl 0212.26101 · Zbl 0212.26101
[7] R. Harvey and R. O. Wells Jr. , Compact holomorphically convex subsets of a Stein manifold , Trans. Amer. Math. Soc. 136 ( 1969 ), 509-516. MR 38 #3470 | Zbl 0175.37204 · Zbl 0175.37204
[8] H. Hironaka , Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II , Ann. of Math. 79 ( 1964 ), 109-326. MR 33 #7333 | Zbl 0122.38603 · Zbl 0122.38603
[9] C. Houzel , Géométrie analytique locale , Séminaire Henri Cartan, 13ème année : 1960 / 1961 , Ecole Normale Supérieure, 1962 . Numdam | Zbl 0121.15906 · Zbl 0121.15906
[10] * M. Kashiwara and T. Kawai , Pseudo-differential operators in the theory of hyperfunctions , Proc. Japan Acad. 46 ( 1970 ), 1130-1134. Article | MR 47 #4295 | Zbl 0229.47034 · Zbl 0229.47034
[11] B. Malgrange , Cohomologie de Spencer (d’après Quillen) .
[12] V. P. Palamodov , Differential operators in the class of convergent power series, and the Weierstrass preparation theorem , Funkcional. Anal. i Prilozen 2 ( 1968 ), 58-69. MR 38 #4795 | Zbl 0176.45701 · Zbl 0176.45701
[13] V. P. Palamodov , Linear differential operators with constant coefficients , Grundlehren der mathematischen Wissenschaften 168, Springer, 1970 . MR 41 #8793 | Zbl 0191.43401 · Zbl 0191.43401
[14] D. Quillen , Formal properties of overdetermined system of linear partial differential equations , Thesis ( 1964 ). · Zbl 1295.35005
[15] M. Sato , Theory of hyperfunctions I, II , Journ. Fac. Sci. Univ. Tokyo Sect. I 8 ( 1959 ), 139-193; ibid. 8 ( 1960 ), 387-437. MR 24 #A2237 | Zbl 0087.31402 · Zbl 0087.31402
[16] M. Sato , Hyperfunctions and partial differential equations , Proceedings of the International Conference on Functional Analysis and Related Topics (Tokyo, April, 1969), University of Tokyo Press, Tokyo, 1970 . Zbl 0208.35801 · Zbl 0208.35801
[17] * T. Kawai and H. Komatsu , Boundary values of hyperfunction solutions of linear partial differential equations , Publ. Res. Inst. Math. Sci. 7 ( 1971 / 1972 ), 95-104. Zbl 0225.35032 · Zbl 0225.35032
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