 
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 pathconnected, semilocally simply connected, locally compact,
secondcountable, and Hausdorff. Unless otherwise specified, they are also pathconnected.
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 squareroot sheaf on X.
(a) Let f : X X be the map f(z) = z2
. Identify fQ and f1
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
