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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.

35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
32C38 Sheaves of differential operators and their modules, \(D\)-modules
Full Text: DOI
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