Extension of functors for algebras of formal deformation

Details Extension of functors for algebras of formal deformation Extension of functors for algebras of formal deformation

2012-01-02

arxiv.org/abs/1201.0536v2

Mathematics

Algebraic Geometry

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Scientific paper

Given two complex manifolds $X$ and $Y$ and a right exact functor between subcategories $\mathcal{S}$ and $\mathcal{S}'$ of modules over an algebra of formal deformation in the sense of Kashiwara and Schapira, respectively on $X$ and on $Y$, we give conditions on $\mathcal{S}, \mathcal{S}'$ for the existence of a canonical extension to the subcategory of modules such that the cohomology associated to the action of the formal parameter $\hbar$ belongs to $\mathcal{S}$. We give an explicit construction and prove that when the initial functor is exact so is its extension. We apply our construction to give a meaning to inverse and direct image with respect to a morphism, Fourier transform, specialization and microlocalization along a submanifold, nearby and vanishing cycles along an hypersurface in the framework of $\shd[[\hbar]]$-modules. We also obtain a Cauchy-Kowalewskaia-Kashiwara theorem in the non-characteristic case as well as comparison theorems for regular holonomic $\shd[[\hbar]]$-modules and a coherency criterion for proper direct images of good $\shd[[\hbar]]$-modules.

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