**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 --> C_{1} --> C_{0} --> 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

L_{0}F(M) = F(M) = M/nM.

L_{1}F(M) = M[n] = {x in M such that nx=0}

L_{n}F(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 L_{0}F(M)=FM and
L_{n}F(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 L_{0}F is naturally equivalent to F.