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 functors **

Let be a map of spaces. We have defined the functor

earlier, such that consists of the sections of whose support is proper over ; is always a subsheaf of , equal to it if is proper. When is a point, we get the functor

One can check that is in fact a sheaf. The observation here is that a map of topological spaces is proper if and only if there is an open cover of such that is proper for each . Now 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 , 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 . (more…)