Let $i: M\hookrightarrow X$ be the inclusion of a closed submanifold in a smooth manifold $X$. I denote by $T_MX$ the normal bundle to $M$ in $X$, by $T^{\ast}_MX$ its dual bundle, and by $D^b(X)$ the derived category of the category of sheaves of $A$-modules on $X$, where $A$ is a unital ring.
For $\mathcal{F}\in D^b(X)$, we define the microlocalization of $\mathcal{F}$ along $M$, written $\mu_M(\mathcal{F})\in D^b(T^{\ast}_MX)$, as the Fourier-Sato transform of $\nu_M(\mathcal{F})\in D^b(T_MX)$, i.e. the specialization of $\mathcal{F}$ along $M$.
I won't report here the precise definition of the two things mentioned above (which can be found in Kashiwara and Schapira's Sheaves on Manifolds), since it would be too long to write and probably useless: as far as I understand, $\nu_M(\mathcal{F})$ should be a sheaf that keeps track of the local behaviour of $\mathcal{F}$ near $M$, while the Fourier-Sato transform is an equivalence of categories $D^b(T_MX)\rightarrow D^b(T^{\ast}_MX)$ that is useful mainly because
- it's always better to work in the cotangent bundle (e.g. it has a symplectic structure)
- it seems to outline the points around which $\mathcal{F}$ is badly behaved (I have sketched the calculation of the Fourier transform of the kernel of the morphism of sheaves $P : \mathcal{O}_{\mathbb{R}}\rightarrow\mathcal{O}_{\mathbb{R}}$ given by the differential operator $P = x\frac{\partial}{\partial x} - \alpha$, $\alpha\in\mathbb{R}$, with some colleague and it seems to be a complex concentrated in degree 1, where the cohomology is a skyscraper sheaf at the point $0\in\mathbb{R}$).
I'm not completely sure about my intuition on these things, so I'd appreciate if somebody would point out some possible mistake.
I am currently reading Kashiwara and Schapira's Sheaves on Manifolds, and right after the definition of the microlocalization, a theorem (4.3.2) containing a long list of properties of $\mu_M(\mathcal{F})$ is stated without proof, and it is claimed to be a consequence of Theorem 4.2.3 (which in fact at first glance looks like an analogous list of properties of the specialization functor) and other results about the Fourier-Sato transform. I'm trying to understand how this proof should go, and in particular I am mostly interested in the proof of part (ii). What I thought is the following: by Proposition 3.7.12 (ii), for any $V\subseteq T^{\ast}_MX$ open convex cone, $$R\Gamma(V, \mu_M(\mathcal{F}) \cong R\Gamma_{V^°}(T_MX, \nu_M(\mathcal{F}))$$$$R\Gamma(V, \mu_M(\mathcal{F})) \cong R\Gamma_{V^°}(T_MX, \nu_M(\mathcal{F}))$$ where by the right-hand side I mean derived section supported in the polar set of $V$, defined by $$V^° = \{y\in T_MX | \pi(y)\in\tau(V)\ \text{and} \langle y,x\rangle\geq 0 \forall x\in\tau^{-1}\pi(y)\cap V\}$$ where $\tau$ and $\pi$ are the obvious bundle maps. Then I'd want to use Theorem 4.2.3 part (iii) to get the thesis, but the problem is that $V^°$, which is indeed conic, has no reason to be closed (actually I might even have counterexamples for that)...
Summarizing: I'm looking for a proof of Theorem 4.3.2 in Sheaves on Manifolds, but I would also be satisfied just with the proof of part (ii).
