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].

For the entire collection see [Zbl 1195.18001].

Reviewer: Rémi Arcadias (Tokyo)

##### 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 |