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 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 Collections: Mathematics