Summary: Applications of Homological Algebra Introduction to Perverse Sheaves
Spring 2007 P. Achar
Problem Set 7
March 13, 2007
1. Let F be a sheaf on X, and let G be an injective sheaf on X. Show that Hom(F, G) is flasque. Your
proof should be valid in the category of sheaves of abelian groups (in particular, do not assume that
F is flat, and do not cite the result proved in class that states that Hom(F, G) is injective if F is flat
and G is injective).
2. Show that a sheaf F is injective if and only if Hom(-, F) is an exact functor.
3. (Uniqueness of adjoint functors) Let F : A B be an additive functor of abelian categories, and let
G, H : B A be two functors that are both right adjoint to F. Show that G H. I.e., show that
there is a rule that assigns to every object B B a morphism (B) : G(B) H(B) in A such that
for every morphism f : B C in B, the following square commutes: