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,
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.