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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

  

Source: Achar, Pramod - Department of Mathematics, Louisiana State University

 

Collections: Mathematics