Math 721
Spring 2001
Homework 2
a)
Show
that if are onto and
is one to one, then
is onto.
b)
Show
that if are one to one and
is onto, then
is one to one.
c)
Conclude
that if are all isomorphisms,
then so is
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.
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
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.
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)