Summary: Applications of Homological Algebra Introduction to Perverse Sheaves
Spring 2007 P. Achar
Derived Categories and Derived Functors
March 6, 2007
Recall that distinguished triangles in K(A) have four important properties. These properties are essential
for even showing that D(A) is a well-defined category: specifically, composition of morphisms in D(A) is
not an obvious operation, and we have to use distinguished triangles in K(A) to show how to do it.
Theorem 1. Distinguished triangles in D(A) enjoy the following properties:
(0) (Existence) Any morphism A·
can be completed to a distinguished triangle A·
(14) The Identity, Rotation, Completion, and Octahedral properties of distinguished triangles in K(A).
Proposition 2. A distinguished triangle A·