 
Summary: Applications of Homological Algebra Introduction to Perverse Sheaves
Spring 2007 P. Achar
Problem Set 11
April 12, 2007
In all of the following problems, X is a stratified space with stratification S, and p : S Z is a perversity
function.
1. Prove that Db
c(X) is a triangulated category. (All the axioms except one are obvious because they
hold in Db
(X). The only thing to show is that given a morphism f : F G, you can extend it to a
distinguished triangle F G H F[1]. You can of course do that in Db
(X), but is H necessarily
constructible?) (Hint: First reduce the problem to the case of one stratum. In that case, be careful:
you are dealing with sheaves whose cohomology sheaves are locally constant, but you cannot assume
that the sheaves themselves are complexes of locally constant ordinary sheaves.)
2. Let S be a stratum, E a local system on S, and F a perverse sheaf with support contained in S S.
Show that
Hom(IC(S, E), F) = Hom(F, IC(S, E)) = 0.
For the remaining problems, assume that p is a GoreskyMacPherson perversity. That is, p(S) = ~p(dim S),
where ~p : N Z is a function satisfying
