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 9 March 22, 2007 In this problem set, you will show that the derived category of the heart of a t-structure on a triangulated category need not be equivalent to the original triangulated category. Let X be the 2-sphere. 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 t-structure obtained by restricting the standard t-structure on Db (ShX) to C: C0 Collections: Mathematics