 
Summary: Applications of Homological Algebra Introduction to Perverse Sheaves
Spring 2007 P. Achar
Proper PullBack and PoincarŽeVerdier Duality
March 13, 2007
Convention. We now add an assumption to our standing list of assumptions on topological spaces. For
each topological space X, we assume there is a number r such that for any exact sequence
F0
· · · Fr
0
of sheaves on X, if F0
, . . . , Fr1
are all soft, then so is Fr
. (This assumption is true for manifolds and for
closed subsets of manifolds.)
We also add a hypothesis on sheaves. From now on, all sheaves will be assumed to be sheaves of finite
dimensional vector spaces, and ShX will denote the category of such sheaves.
The main result in this set of notes is the following.
Theorem 1. Let f : X Y be a continuous map. There is a functor f!
: D+
(ShY ) D+
