 
Summary: Applications of Homological Algebra Introduction to Perverse Sheaves
Spring 2007 P. Achar
Problem Set 10
March 29, 2007
1. A functor F : C1 C2 between two triangulated categories with tstructures is said to be texact if
F(C0
1 ) C0
2 and F(C0
1 ) C0
2 . Let DU , DZ, and D be categories of sheaves as in the theorem
on gluing of tstructures. Show that the tstructure on D described in that theorem is the unique
tstructure on D such that Ri : DZ D and j1
: D DU are texact functors.
2. Suppose we give DU and DZ the standard tstructure. Show that the tstructure on D described by the
gluing theorem is the standard tstructure. Also, show that the middleextension functor j! coincides
in this case with the (nonderived) extensionbyzero functor j!.
3. Suppose DU has the standard tstructure. By shifting the standard tstructure on DZ and then gluing,
can you obtain a tstructure on D for which the middleextension functor coincides with the nonderived
pushforward j? How about Rj? Rj!? (Hint: The answers for j and Rj! are "yes." For Rj, it
depends on properties of the topological space U. You should find a condition on U under which the
