## 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

### Keywords:

M. Kashiwara’s thesis; $$\mathcal D$$-modules
Full Text:

### References:

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