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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

  

Source: Achar, Pramod - Department of Mathematics, Louisiana State University

 

Collections: Mathematics