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 10 March 29, 2007 1. A functor F : C1 C2 between two triangulated categories with t-structures is said to be t-exact 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 t-structures. Show that the t-structure on D described in that theorem is the unique t-structure on D such that Ri : DZ D and j-1 : D DU are t-exact functors. 2. Suppose we give DU and DZ the standard t-structure. Show that the t-structure on D described by the gluing theorem is the standard t-structure. Also, show that the middle-extension functor j! coincides in this case with the (non-derived) extension-by-zero functor j!. 3. Suppose DU has the standard t-structure. By shifting the standard t-structure on DZ and then gluing, can you obtain a t-structure on D for which the middle-extension functor coincides with the non-derived push-forward 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 Collections: Mathematics