Exercises on Derived Categories and Perverse Sheaves PRAMOD N. ACHAR Summary: Exercises on Derived Categories and Perverse Sheaves PRAMOD N. ACHAR Let O = C[x]. Regard this as a graded ring by putting deg x = 1. All O-modules below should be assumed to be graded. In particular, Db (O) will denote the bounded derived category of graded O-modules. For any O-modules M, we write M(n) for the same module with a shift in grading by n. Thus, O(n) is the free O-module generated by a generator in degree -n. Of course, O-modules are the same as quasicoherent sheaves over A1 . The restriction of an O-module M to the open set U = A1 {0} is denoted MU . In particular, we have OU = C[x, x-1 ]. Lecture 1: Basics of derived categories 1. Let D be a triangulated category that is also abelian, and in which all distinguished triangles are short exact sequences. Prove that D contains only the zero object. 2. Let M = O/(x). Check that RHom(M, O(-1)[1]) M. 3. Let A be an abelian category with enough projectives (or enough injectives). Show that the following conditions are equivalent: (a) Ext2 Collections: Mathematics