dg-Methods for Microlocalization

  • Stéphane Guillermou

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


For a complex manifold XX the ring of microdifferential operators EX\mathcal{E}_X acts on the microlocalization μhom(F,OX)\mu hom(F,\mathcal{O}_X) for FF in the derived category of sheaves on XX. Kashiwara, Schapira, Ivorra, Waschkies proved as a byproduct of their new microlocalization functor for ind-sheaves, μX\mu_X, that μhom(F,OX)\mu hom(F,\mathcal{O}_X) can in fact be defined as an object of D(EX)\mathrm{D}(\mathcal{E}_X): this follows from the fact that μXOX\mu_X \mathcal{O}_X is concentrated in one degree.
In this paper we prove that the tempered microlocalization Tμhom(F,OX)T - \mu hom(F,\mathcal{O}_X) and in fact μXOXt\mu_X \mathcal{O}_X^t also are objects of D(EX)\mathrm{D}(\mathcal{E}_X). Since we don't know whether μXOXt\mu_X \mathcal{O}_X^t is concentrated in one degree we built resolutions of EX\mathcal{E}_X and μXOXt\mu_X \mathcal{O}_X^t such that the action of EX\mathcal{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