Let $X$ be a (finite-dimensional) topological manifold and suppose $i: Z \to X$ is the inclusion of a closed subspace. Then there are several derived functors on derived categories of sheaves of abelian groups associated to $i$:
- The pushforward $i_{\ast}: Sh(Z) \to Sh(X)$, equal in this case to the proper pushforward $i_{!}$ which is an exact functor;
- The inverse image $i^{-1}: Sh(X) \to Sh(Z)$, also an exact functor;
- The functor $i^{!}: D^b Sh(X) \to D^b Sh(Z)$, adjoint to $i_{\ast} = i_{!}: D^b Sh(Z) \to D^b Sh(X)$.
Both of the following statements seem to be true in this context, but I have been unable to find them stated anywhere, which makes me feel somewhat suspicious.
- Since $i^{-1}$ is adjoint to $i_{\ast}$ on derived categories of sheaves, we must have $i^{!} = i^{-1} : D^b Sh(X) \to D^b Sh(Z)$;
- By an analog of `Kashiwara's equivalence' for $D$-modules, the functors $i_{\ast}$ and $i^{-1}$ are inverses should give an equivalence of categories between $Sh(Z)$ and $Sh_Z(X)$, the subcategory of sheaves on $X$ with support in $Z$.
Is someone able to provide a reference for these statements in the literature, or provide a counterexample to either?
