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…)