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Triangulated categories for the analysts. (English) Zbl 1213.18008
Holm, Thorsten (ed.) et al., Triangulated categories. Based on a workshop, Leeds, UK, August 2006. Cambridge: Cambridge University Press (ISBN 978-0-521-74431-7/pbk). London Mathematical Society Lecture Note Series 375, 371-388 (2010).
The aim of that paper is to review how the tools of algebraic geometry, namely derived categories of sheaves, apply to analysis. First, the classical functional spaces on a complex manifold $$X$$ are replaced by functorial spaces $$R\mathcal{H}\mathrm{om}(G,\mathcal{O}_X)$$ called generalized functions, where $$G$$ is an object of the derived category of $$\mathbb{R}$$-constructible sheaves on the real underlying manifold to $$X$$. As an example, the theory of Sato’s hyperfunctions (c.f. M. Sato [J. Fac. Sci., Univ. Tokyo, Sect. I 8, 139–193 (1959; Zbl 0087.31402) and J. Fac. Sci. Univ. Tokyo, Sect. I 8, 387–437 (1960; Zbl 0097.31404)]) is described. After having briefly introduced the theory of $$\mathcal{D}$$-modules, which represent general systems of linear partial differential equations, the author explains that the space of solutions of a $$\mathcal{D}$$-module $$\mathcal{M}$$ is replaced by a complex of sheaves of vector spaces $$R\mathcal{H}\mathrm{om}(G,F)$$, where $$F=R\mathcal{H}\mathrm{om}_{\mathcal{D}_X}(\mathcal{M},\mathcal{O}_X)$$. In order to study these complexes of sheaves, the notion of microsupport, a closed conic subset of $$T^{\star}X$$, is defined, and an application to elliptic systems is given. That latter notion leads the author to review some constructions from microlocal analysis: Fourier-Sato transform, specialization and microlocalization, and microdifferential operators (first constructed in M. Sato, T. Kawai and M. Kashiwara [Lect. Notes Math. 287, 263–529 (1973; Zbl 0277.46039)]). A detailed reference for microlocal analysis is M. Kashiwara and P. Schapira [“Sheaves on manifolds”, Berlin etc.: Springer-Verlag (1990; Zbl 0709.18001)]. Finally, following M. Kashiwara and P. Schapira [“Ind-sheaves”, Paris: Société Mathématique de France (2001; Zbl 0993.32009)], the use of Grothendieck topologies is suggested in order to treat generalized functions with growth conditions.
For the entire collection see [Zbl 1195.18001].
##### MSC:
 18E30 Derived categories, triangulated categories (MSC2010) 35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs 32A45 Hyperfunctions 32C38 Sheaves of differential operators and their modules, $$D$$-modules