Summary: Applications of Homological Algebra Introduction to Perverse Sheaves
Spring 2007 P. Achar
Problem Set 5
February 15, 2007
1. Gelfand & Manin, Methods of Homological Algebra, Exercise II.5.2 (page 119).
2. Ibid., Exercise II.5.5 (page 120). (Note: See Exercise II.5.4 for the definition of "f(y).")
3. Let A· f
- B· g
be a distinguished triangle (in K(A) or D(A)). Show that g f = 0.
4. Let 0 - A· f
- B· g
- 0 be a short exact sequence in C(A). Define a morphism :
by the matrix 0 g . Show that is indeed a morphism of complexes (i.e., that it
commutes with differentials) and that the following diagram commutes: