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Math 5125
Monday, November 28
November 28, Ungraded Homework
Exercise 10.5.9 on page 404
Assume that
R
is commutative with a 1.
(a) Prove that the tensor product of two free
R
modules is free.
(b) Use (a) to prove that the tensor product of two projective
R
modules is projective.
(a) First note that
R
⊗
R
∼
=
R
as
R
modules. Here is a sketch proof: we have
R
maps
f
:
R
⊗
R
R
→
R
satisfying
f
(
r
⊗
s
) =
rs
, and
g
:
R
→
R
⊗
R
R
deﬁned by
g
(
r
) =
r
⊗
1. Then
fg
=
g f
=
1, which shows that
f
and
g
are isomorphisms. Using the facts
(
A
⊕
B
)
⊗
R
C
∼
= (
A
⊗
R
C
)
⊕
(
B
⊗
R
C
)
A
⊗
R
(
B
⊕
C
)
∼
= (
A
⊗
R
B
)
⊕
(
A
⊗
R
C
)
,
we deduce by induction that
R
m
⊗
R
R
n
∼
=
R
mn
. This proves that the tensor product of
free modules is free in the ﬁnitely generated case. Actually this argument is still OK if
the modules are not ﬁnitely generated.
(b) Suppose
P
,
Q
are projective
R
modules. Then we may write
P
⊕
A
=
E
and
Q
⊕
B
=
F
for some
R
modules
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 Fall '07
 PALinnell
 Math, Algebra

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