# dg-Methods for Microlocalization

### Stéphane Guillermou

Université de Grenoble I, Saint-Martin-d'Hères, France

## Abstract

For a complex manifold $X$ the ring of microdifferential operators $E_{X}$ acts on the microlocalization $μhom(F,O_{X})$ for $F$ in the derived category of sheaves on $X$. Kashiwara, Schapira, Ivorra, Waschkies proved as a byproduct of their new microlocalization functor for ind-sheaves, $μ_{X}$, that $μhom(F,O_{X})$ can in fact be defined as an object of $D(E_{X})$: this follows from the fact that $μ_{X}O_{X}$ is concentrated in one degree.

In this paper we prove that the tempered microlocalization $T−μhom(F,O_{X})$ and in fact $μ_{X}O_{X}$ also are objects of $D(E_{X})$. Since we don't know whether $μ_{X}O_{X}$ is concentrated in one degree we built resolutions of $E_{X}$ and $μ_{X}O_{X}$ such that the action of $E_{X}$ 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. 99–140

DOI 10.2977/PRIMS/32