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The Cauchy-Kowalevski theorem for $$\mathcal E$$-modules. (English) Zbl 0906.35004
The author proves a version of the Cauchy-Kowalevski theorem for $${\mathcal E}_X$$-modules and for holomorphic functions in the category $$D^b(Y;p_Y).$$ Namely, upon considering a complex analytic manifold $$X,$$ a point $$p_X\in T^*X$$ and a coherent left $${\mathcal E}_X$$-module $${\mathcal M},$$ he proves that the object $$R{\mathcal H}om_{{\mathcal E}_X}({\mathcal M},{\mathcal O}_X)_{p_X}$$ is well defined in $$D^b(X;p_X)$$ (by proving, among other properties that, for instance, it depends only on $${\mathcal M}$$ up to isomorphisms, and that it is functorial with respect to $${\mathcal M}).$$ He next takes a morphism $$f\colon Y\rightarrow X$$ of complex manifolds, a point $$p\in Y\times_XT^*X,$$ considers $$p_X=f_\pi(p)$$ and $$p_Y= ^t f'(p),$$ and supposes that $${\mathcal M}$$ is defined in a neighborhood of $$p_X$$ and non-characteristic with respect to $$f.$$ Upon denoting by $$\underline{f_p}^{-1}{\mathcal M}$$ the inverse image of $${\mathcal M}$$ by $$f,$$ he constructs the natural morphism $$f_p^{-1}R{\mathcal H}om_{{\mathcal E}_X}({\mathcal M},{\mathcal O}_X)_{p_X}\rightarrow R{\mathcal H}om_{{\mathcal E}_Y}(\underline{f_p}^{-1}{\mathcal M},{\mathcal O}_Y)_{p_Y}$$ in the category $$D^b(Y;p_Y)$$ and proves that it is an isomorphism. In the final section, he gives an application.

##### MSC:
 35A10 Cauchy-Kovalevskaya theorems 32C38 Sheaves of differential operators and their modules, $$D$$-modules 58J15 Relations of PDEs on manifolds with hyperfunctions 32A10 Holomorphic functions of several complex variables
##### Keywords:
$${\mathcal E}_X$$-modules
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##### References:
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