This is the third in a series of posts started here (in particular, the notation is kept from there) intended to cover the basics of Verdier duality. Here, I will discuss the lower shriek functors needed even to state Verdier duality (in the most general form, at least); as we will see, the class of soft sheaves will be acyclic with respect to this functor. To see this, though, we shall need to prove some general facts on how push-forward behave with respect to base change, which are themselves of independent interest.

1. The ${f_!}$ functors

Let ${f: X \rightarrow Y}$ be a map of spaces. We have defined the functor $\displaystyle f_! : \mathbf{Sh}(X) \rightarrow \mathbf{Sh}(Y)$

earlier, such that ${f_!(U) }$ consists of the sections of ${\mathcal{F}(f^{-1}(U))}$ whose support is proper over ${U}$ ; ${f_!\mathcal{F} }$ is always a subsheaf of ${f_*\mathcal{F}}$, equal to it if ${f}$ is proper. When ${Y}$ is a point, we get the functor $\displaystyle \mathcal{F} \mapsto \Gamma_c(X, \mathcal{F}) = \left\{\text{global sections with proper support}\right\}.$

One can check that ${f_! \mathcal{F}}$ is in fact a sheaf. The observation here is that a map ${A \rightarrow B}$ of topological spaces is proper if and only if there is an open cover ${\left\{B_i\right\}}$ of ${B}$ such that ${A \times_B B_i \rightarrow B_i}$ is proper for each ${i}$. Now ${f_!}$ is a left-exact functor, as one easily sees. We now want to show that the class of soft sheaves is acyclic with respect to ${f_!}$, and in particular so that one may use soft resolutions to compute the derived functors. To do this, we shall prove a general “base change” theorem that will compute the stalk of ${f_! \mathcal{F}}$. (more…)