Applications of Homological Algebra Introduction to Perverse Sheaves Spring 2007 P. Achar 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proof should be valid in the category of sheaves of abelian groups (in particular, do not assume that F is flat, and do not cite the result proved in class that states that Hom(F, G) is injective if F is flat and G is injective). 2.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: G(B) (B) // G(f) H(B) H(f) Collections: Mathematics