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I proved part (i) of the following:

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Can you tell me if my proof of $(i)\Longleftarrow$ is correct? Here goes:

Let $M^\prime$ and $M^{\prime \prime}$ be Noetherian. Let $L_n$ be an ascending chain of submodules in $M$. Then $\alpha^{-1}(L_n)$ is an ascending chain of submodules in $M^\prime$. Hence $\alpha^{-1}(L_n)$ is stationary, that is, $\alpha^{-1}(L_n) = \alpha^{-1}(L_{n+1})$ for $n$ large enough. Hence $L_n = L_{n+1}$ for $n$ large enough since $\alpha$ is injective hence $L_n$ is stationary.

The proof given in Atiyah-Macdonald is the following but I don't understand why they need both maps, $\alpha$ and $\beta$:

enter image description here

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Wow you do ask a lot of questions on commutative algebra..... –  user38268 Jul 20 '12 at 14:55

1 Answer 1

up vote 6 down vote accepted

If this proof worked then we would be able to conclude that a module is Noetherian if it has a Noetherian submodule. This is certainly not the case — as a silly example, the trivial submodule is always Noetherian. If we forget about $\beta$ then it is true that $(\alpha^{-1}(L_n))_n$ will stabilize, but it's not as if $\alpha(\alpha^{-1}(L_n)) = L_n$, since $\alpha$ need not be surjective.

To use Atiyah and MacDonald's proof, you want to show that if $N_1 \subset N_2$ are submodules of $M$ such that $\alpha^{-1}(N_1) = \alpha^{-1}(N_2)$ and $\beta(N_1) = \beta(N_2)$, then $N_1 = N_2$. For this, take $x \in N_2$. By assumption there is a $y \in N_1$ such that $\beta(y) = \beta(x)$. By exactness, then, there is a $z \in M'$ such that $\alpha(z) = x - y \in N_2$. Thus $z \in \alpha^{-1}(N_2) = \alpha^{-1}(N_1)$. Do you see how to conclude that $x \in N_1$?

[We do need that $N_1 \subset N_2$, by the way. Think about $M = \mathbb R^2$ with $M' = \text{the $x$-axis}$ and the family of lines $y = mx$ for $m \neq 0$.]

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From $\alpha^{-1}(x-y) \in \alpha^{-1}(N_2)$ it follows that $x-y \in \alpha(\alpha^{-1}(N_2)) \subset N_2$ and since by assumption, $y \in N_2$, we may conclude that $x \in N_2$. Is that correct? –  Rudy the Reindeer Jul 21 '12 at 12:02
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@Clark Looks good to me! –  Dylan Moreland Jul 21 '12 at 16:35
    
Cool, thank you very much! –  Rudy the Reindeer Jul 21 '12 at 16:43

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