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 4 February 8, 2007 1. Let p : Y X be a finite-to-one covering map. If E is a local system on Y , show that pE is a local system on X. If p is not finite-to-one, 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 one-to-one 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 right-exact and that T is left-exact. (Be sure to do this using the language of categories--you 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 Collections: Mathematics