 
Summary: Applications of Homological Algebra Introduction to Perverse Sheaves
Spring 2007 P. Achar
Problem Set 4
February 8, 2007
1. Let p : Y X be a finitetoone covering map. If E is a local system on Y , show that pE is a local
system on X. If p is not finitetoone, the proof of the previous statement does not go through, but it
can be fixed up by imposing an additional condition on E. Find such a condition, and prove that with
this extra condition, pE is again a local system. (Hint: If Y is compact, no additional condition on E
is needed.)
2. Recall that there is a onetoone correspondence between covering spaces over X (up to isomorphism)
and subgroups of its fundamental group 1(X, x0) (up to conjugacy. Suppose p : Y X is a covering
map corresponding to the subgroup H 1(X, x0). Let E = C[1(X, x0)/H], the vector space of
formal linear combinations of cosets of H with complex coefficients. There is an obvious representation
of 1(X, x0) on E. Show that the local system pC is the one corresponding to this representation.
3. Show that the category of sheaves of abelian groups on a fixed topological space X is an abelian
category.
4. Let S : A B and T : B A be an adjoint pair of functors. Show that S is rightexact and that T
is leftexact. (Be sure to do this using the language of categoriesyou cannot talk about "elements"
of kernels and images, because in a general abelian category, the concept of "element" may not make
sense.) It immediately follows that the functors
