 
Summary: Applications of Homological Algebra Introduction to Perverse Sheaves
Spring 2007 P. Achar
Problem Set 9
March 22, 2007
In this problem set, you will show that the derived category of the heart of a tstructure on a triangulated
category need not be equivalent to the original triangulated category.
Let X be the 2sphere. Let C be the full subcategory of Db
(ShX) consisting of complexes of sheaves F all
of whose cohomology sheaves Hi
(F) are constant ordinary sheaves on X.
1. Show that C inherits from Db
(ShX) the structure of a triangulated category. (The main thing to show
is that any morphism can be completed to a distinguished triangle: if F G is a morphism in C, then
of course there is a distinguished triangle F G H F[1] in Db
(ShX), but is H necessarily in C?)
2. Let (C0
, C0
) be the tstructure obtained by restricting the standard tstructure on Db
(ShX) to C:
C0
