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dg-methods for microlocalization. (English) Zbl 1223.35019
Let $$X$$ be a complex manifold, and let $${\mathcal E}_X$$ be the ring of microdifferential operators. It acts on the microlocalization $$\mu \hom(F,{\mathcal O}_X)$$, for $$F$$ in the derived category of sheaves on $$X$$. As a consequence of their new microlocalization, M. Kashiwara, P. Shapira, F. Ivorra and I. Waschkies proved [in: J. Bernstein (ed.) et al., Studies in Lie theory. Dedicated to A. Joseph on his sixtieth birthday. Basel: Birkhäuser. Progress in Mathematics 243, 171–221 (2006; Zbl 1098.35008)] that $$\mu\hom(F,{\mathcal O}_X)$$ can be defined as an object of $$D({\mathcal E}_X)$$ (and this follows from the fact that $$\mu_X{\mathcal O}_X$$ is concentrated in one degree).
In this paper, the author proves that the tempered microlocalization $$T$$-$$\mu \hom(F,{\mathcal O}_X)$$ and $$\mu_X{\mathcal O}_X^t$$ are also objects of $$D({\mathcal E}_X)$$. To accomplish this, since one does not know if $$\mu_X{\mathcal O}_X^t$$ is concentrated in one degree, the author constructs resolutions of $${\mathcal E}_X$$ and $$\mu_X{\mathcal O}_X^t$$ such that the action of $${\mathcal E}_X$$ is realized in the category of complexes, and then builds a microlocalization in this framework.

##### MSC:
 35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs 32C38 Sheaves of differential operators and their modules, $$D$$-modules
##### Keywords:
microdifferential operators
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##### References:
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