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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].
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