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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)).
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