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Summary: Applications of Homological Algebra Introduction to Perverse Sheaves
Spring 2007 P. Achar
Problem Set 1
1. Prove the following statements (I did some of these in class). F and G are sheaves unless otherwise
specified.
(a) Let {Vi} be an open cover of U, and let s, t F(U). If s|Vi
= t|Vi
for all i, then s = t. (In
particular, if s|Vi
= 0 for all i, then s = 0.)
(b) A section is determined by its germs. That is, if s, t F(U) are sections such that sx = tx for all
x U, then s = t.
(c) Suppose F is presheaf, G is a sheaf, and F G. Define a subsheaf F G by
F (U) = {s G(U) | there is a covering {Vi} of U such that s|Vi
F(Vi) for all i}.
Then F+
F . In particular, if F is a sheaf, then F+
F.
(d) Let F be a presheaf. The stalks of F+
are isomorphic to those of F.
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