Sheaves on subanalytic sites. (English) Zbl 1171.32002

Prelli, Luca

Rend. Semin. Mat. Univ. Padova 120, 167-216 (2008).

Let \(k\) be a field and \(X\) be a real analytic manifold. The spaces of functions which are not defined by local properties do not define sheaves on \(X\), they define sheaves on a site associated to \(X\), the subanalytic site \(X_{sa}\).
M. Kashiwara and P. Schapira [Ind-sheaves. Astérisque. 271. Paris: Société Mathématique de France (2001; Zbl 0993.32009)] introduced the notion of ind-sheaves and defined the so-called six Grothendieck operations in this framework. Restriction to \(\mathbb{R}\)-constructible sheaves gives an equivalence between the category of ind \(\mathbb{R}\)-constructible sheaves on \(X\) and the categroy \(\text{Mod}(k_{X_{sa}})\) of sheaves on the subanalytic site associated to \(X\).
The paper gives a direct, self-contained construction of the six Grothendieck operations on \(\text{Mod}(k_{X_{sa}})\) without using the theory of ind-sheaves.

Reviewer: Gerhard Pfister (Kaiserslautern)

MSC:

32B99	Local analytic geometry
14P05	Real algebraic sets
14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)
35C05	Solutions to PDEs in closed form
32S99	Complex singularities

Keywords:

subanalytic site; ind-sheaf; Grothendieck operation

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