Advanced Algebra I
Homework 15 due on Jan. 19, 2007
(1) Complete the proof of Theorem 4.5.6 and 4.5.7
(2) Let F : A → B be a left exact functor and A has enough injectives. Show that RiF (A) := Hi(F (I•)) is well-defined.
That is, independent of choice of injective resolution I•.
(3) Given a complex K•, construct an injective resolution I• of K•. That is, a quasi-isomorphism f : K• → I•.
(4) In the category of abelian groups, show that an injective object is a divisible group.
(5) We can define projective in a similar way (with arrow reversing ). That is for any exact sequence B → C → 0 and f : P → C,α there exist g : P → B such that gα = f .
Show that for any exact sequence 0 → A → B → P → 0 with P being projective, the sequence splits.
We now consider the category of abelian groups Ab. Deter- mine the projective objects. (Hint: every abelian group is a quotient of free abelian group.)
1
