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 3 February 1, 2007 In this problem set (and henceforth in the course), the following slight abuse of language will be made: if : 1(X, x0) GL(E) is a representation of 1(X, x0) on the vector space E, we will call E itself "the representation." (Thus, "Let E be a representation of 1(X, x0)" means "Let E be a complex vector space, and suppose there is a representation 1(X, x0) GL(E) of 1(X, x0) on E.") 1. Let F, G, and H be sheaves of abelian groups on X. Prove that Hom(F G, H) Hom(F, Hom(G, H)) and Hom(F G, H) Hom(F, Hom(G, H)) by using the corresponding facts for abelian groups. 2. Problem 2 of Problem Set 2 asked you to show that (j!, j-1 ) is an adjoint pair, where j : U X is an open inclusion. State and prove a sheaf-Hom version of that theorem. (Note that it does not make sense to say HomX(j!F, G) HomU (F, j-1 G).) 3. Show that there is an equivalence of categories (local systems on X) (representations of 1(X, x0)). Collections: Mathematics