EXTENSION OF FUNCTORS FOR ALGEBRAS OF FORMAL DEFORMATION

Abstract

AbstractSuppose we are given complex manifoldsXandYtogether with substacks$\mathcal{S}$and$\mathcal{S}'$of modules over algebras of formal deformation$\mathcal{A}$onXand$\mathcal{A}'$onY, respectively. Also, suppose we are given a functor Φ from the category of open subsets ofXto the category of open subsets ofYtogether with a functorFof prestacks from$\mathcal{S}$to$\mathcal{S}'\circ\Phi$. Then we give conditions for the existence of a canonical functor, extension ofFto the category of coherent$\mathcal{A}$-modules such that the cohomology associated to the action of the formal parameter$\hbar$takes values in$\mathcal{S}$. We give an explicit construction and prove that when the initial functorFis exact on each open subset, so is its extension. Our construction permits to extend the functors of inverse image, Fourier transform, specialisation and micro-localisation, nearby and vanishing cycles in the framework of$\mathcal{D}[[\hbar]]$-modules. We also obtain the Cauchy–Kowalewskaia–Kashiwara theorem in the non-characteristic case as well as comparison theorems for regular holonomic$\mathcal{D}[[\hbar]]$-modules and a coherency criterion for proper direct images of good$\mathcal{D}[[\hbar]]$-modules.