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 2 January 30, 2007 From this problem set on, the following assumptions are in effect: · All sheaves are sheaves of complex vector spaces unless otherwise specified. · All topological spaces are locally path-connected, semilocally simply connected, locally compact, second-countable, and Hausdorff. Unless otherwise specified, they are also path-connected. In problems that ask you to "identify" a sheaf, you should either show that the sheaf is isomorphic to some sheaf we have discussed in class, or give as explicit a description as you can of sections of the sheaf over a typical connected open set. 1. Let X = C {0}, and let Q be the square-root sheaf on X. (a) Let f : X X be the map f(z) = z2 . Identify fQ and f-1 Q. (b) Show that Hom(Q, C) = 0. Identify the sheaf Hom(Q, C). (It's not the zero sheaf.) (c) Identify Q Q. Also, show explicitly in this example that the presheaf tensor product Q ps Q is not a sheaf. (d) Let g : X X be the map g(z) = z3 . Identify gC. (Hint: The answer is C F G, where F is Collections: Mathematics