For a complex manifold the ring of microdifferential operators acts on the microlocalization for in the derived category of sheaves on . Kashiwara, Schapira, Ivorra, Waschkies proved as a byproduct of their new microlocalization functor for ind-sheaves, , that can in fact be defined as an object of : this follows from the fact that is concentrated in one degree.
In this paper we prove that the tempered microlocalization and in fact also are objects of . Since we don't know whether is concentrated in one degree we built resolutions of and such that the action of is realized in the category of complexes (and not only up to homotopy). To define these resolutions we introduce a version of the de Rham algebra on the subanalytic site which is quasi-injective. We prove that some standard operations in the derived category of sheaves can be lifted to the (non-derived) category of dg-modules over this de Rham algebra. Then we built the microlocalization in this framework.
Cite this article
Stéphane Guillermou, dg-Methods for Microlocalization. Publ. Res. Inst. Math. Sci. 47 (2011), no. 1, pp. 99–140DOI 10.2977/PRIMS/32