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I am reading Rings and Categories of Modules by Frank W.Aderson,on 124 pages.There is a corollary:

10.3.corollary.If M is finitely cogenerated, then every module that cogenerates M finitely cogennerates M.

there is also a example . The example proves that there is a module $M$ is not finitely generated,but every module cogenerate $m$ finitely generates $M$.Hence the corolaary is right，it can't be a necessary and sufficient conditions.

But in the exercise ,there is a subject:

1。A slight variation of the condition of corollary (10.3) does characterize finitely cogeneraed modules,Prove that $_R$M is finitely cogenerated iff for every module $U$ and every set $A$ ,if there is a finite $F$$\subseteq$$A$ such that $\pi$$_F$$\circ$$f$:$M\longrightarrow U^{F}$ is a monomorphism.[Hint:($\Leftarrow$)Suppose $M_\alpha\leq$$M$ and $\bigcap_AM_\alpha=0$.Set $U=\prod_AM/M_\alpha$ and consider some monomorphism $M\rightarrow U^{A}.$]

I think there is a contradiction between the exercise with the corollary 10.3.Since the exercise states a necessary and sufficient conditions,but the corollary states a necssary contions.In the subject above，give a counter example, Proved if every moudle cogenerates $M$ are all finitely generates $M$,then $M$ is finitely generated.But in the corollary,only state the necessary conditions.Further more there is also have an example to prove the corollary is correct.So I can't found where the problem is .

guojm
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1 Answers1

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If you'll look back at the example we were working on, I think you'll find this is the difference:

In that example, $\exists \phi:\bigoplus_{\mathbb{P}}{\mathbb{Z}_p}\hookrightarrow\prod_ {i\in I}G$ implies $\exists F$ and $\exists\psi:\bigoplus_{\mathbb{P}}{\mathbb{Z}_p}\hookrightarrow \prod_{i\in F}G$. (In fact, $|F|=1$ in that example.)

On the other hand, this modification you are citing says that $\exists F$ such that $\pi_F\circ \phi:\bigoplus_{\mathbb{P}}{\mathbb{Z}_p}\hookrightarrow\prod_{i\in F} G$.

Being able to say that $\phi$ restricts to a monomorphism into finitely many copies is a lot stronger than saying that there exists a completely separate $\psi$ that embeds your module into something else.

rschwieb
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