189-571B: Higher Algebra II

Assignment 10

To be presented: Wednesday, March 28.

The problem marked with a (*) is for everyone to do and to hand in on the day of the problem session. I will return this problem to you on Friday of the same week, so late problems will not be accepted!

If you are stuck on your problem, come see me during my Monday office hours (or make an appointment) to get a hint...

1. (*) a) Show that every finitely generated Z-module M has a free resolution consisting of two non-zero terms:

0 --> C₁ --> C₀ --> M --> 0
(You may use without proof the structure theory of finitely generated modules over a principal ideal domain...)

b) Let n be an integer and let F be the functor from the category of Z-modules to itself which to a module M associates the module M/nM. Show that F is a covariant additive functor in a natural way.

c) Show that the left derived functors of F on a finitely generated module M are given by

L₀F(M) = F(M) = M/nM.
L₁F(M) = M[n] = {x in M such that nx=0}
L_nF(M) =0 for all n>1.

d) Write down concretely the long exact sequence of left derived functors associated to a short exact sequence

0-->M'-->M-->M''-->0
of finitely generated Z-modules, and describe the connecting homomorphism explicitly.

2. (Stanculescu) Check carefully that the definition of L_n(F) given in class is indeed an additive functor from the category of R-modules to the category of abelian groups.

3. (Rua) Prove the following properties of the left derived functors associated to an additive functor on the category of R-modules:

a) If M is projective, then L₀F(M)=FM and L_nF(M)=0.

b) If F is a right exact functor (i.e., if F(M') --> F(M) --> F(M'') -->0 is exact whenever M'-->M-->M''--> 0 is) then L₀F is naturally equivalent to F.