In this post, I’m going to write down the detailed proofs of some of the exercises in Rotman’s Homological Algebra. They were asked in ML and then answered by me.
1. Let
be a torsion abelian group. Then
, where
is the unit circle.
One point is that the structure of the circle group is Now, since
is torsion, then
Therefore, it is enough to show that
We know that functor is the right derived functor of the left exact (covariant) functor
Consider the following exact sequence of abelian groups (injective resolution)
where obviously, the second arrow is inclusion and the third one is the projection.
Applying our left exact (covariant) functor to the above exact sequence, we will obtain the following long exact sequence
Notice that, and
are divisible groups, hence are injective
-modules, thus,
for
Similarly, since is torsion then
Hence,
as desired.
2. Let
be finite abelian groups. Prove that
By the classification of finitely generated abelian groups, we can assume that and
where
are (not necessarily distinct) prime numbers.
Lemma: Let be a finite abelian group, then
Proof of the lemma: Consider the following injective resolution for
where the second arrow is multiplication by and the third one is multiplication by
Applying to the above short exact sequence, we will get
Note that, injectivity of implies that
for
Hence, the required isomorphism. (Notice that the third arrow is the induced multiplication by
from our resolution.)
In particular, setting in the lemma and using the result of the question 1), we get
Note that, and
where
Therefore,
Utilizing the commutativity of and tensor product with (finite) direct sum, and
we obtain the required isomorphism,
3. Let
be a flat left
-module and
be its projective covering, i.e. there is some
-module epimorphism
. Prove that if
is flat, then
is flat.
Rotman’s definition of projective cover of a module is indeed, an ordered pair
where
is projective and
is a surjective morphism with
a superfluous submodule of
I will use the following homological characterization for (left) flat -modules,
By assumption, is a short exact sequence of left
-modules, where
Let be a arbitrary right
-module, then after applying the right exact functor
to the above exact sequence, we will have
By the characterization, for
therefore,
as well as
for
Hence,
is a flat module.
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