 
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 Omodules below should
be assumed to be graded. In particular, Db
(O) will denote the bounded derived category of graded
Omodules.
For any Omodules M, we write M(n) for the same module with a shift in grading by n. Thus, O(n)
is the free Omodule generated by a generator in degree n.
Of course, Omodules are the same as quasicoherent sheaves over A1
. The restriction of an Omodule
M to the open set U = A1
{0} is denoted MU . In particular, we have OU = C[x, x1
].
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
