Math 721

Spring 2001

Homework 2

1. (The Five Lemma) Consider the following commutative diagram with exact rows:

a) Show that if are onto and is one to one, thenis onto.

b) Show that if are one to one and is onto, thenis one to one.

c) Conclude that if are all isomorphisms, then so is

2. Consider the exact sequences:

and

a) Show that the sequence

is also exact.

b) Use part a) to show that any exact sequence can be obtained by “splicing” together short exact sequences.

3. (The snake lemma) Consider the commutative diagram of modules with exact rows:

a) Show that there is an exact sequence:

b) Show that if is one to one, then so is the map

c) Show that if is onto, then so is the map

4. (The 3x3 Lemma) Consider the following commutative diagram

a) Show that if the columns are exact and the bottom two rows are exact, then so is the top.

b) Show that if the columns are exact and the top two rows are exact, then so is the bottom.

5. Given the short exact sequence show that the following conditions are equivalent:

a) There is an module homomorphism such that is the identity map on

b) There is an module homomorphism such that is the identity map on

c)