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I'm reading Commutative Algebra by Bourbaki, and I'm on chapter III of the book. I really think that the author made a typo there, but I'm not so sure. It's Proposition 1, and its Corollary on page 156, and 157 of the book.

Here's what they read:

Proposition 1

Let $A = \bigoplus\limits_{i \in \mathbb{Z}} A_i$ be a graded commutative ring with positive degrees (i.e $A_i = 0, \forall i < 0$), $\mathfrak m$ the graded ideal $\bigoplus\limits_{i \ge 1} A_i$, and $(x_\lambda)_{\lambda \in L}$ a family of homogeneous elements of $A$ of degree $\ge 1$ TFAE:

  1. The ideal of $A$ generated by the family $(x_\lambda)_{\lambda \in L}$ is equal to $\mathfrak{m}$

  2. The family $(x_\lambda)_{\lambda \in L}$ is a system of genereators of the $A_0-$algebra $A$.

  3. For all $i \ge 0$, the $A_0-$module $A_i$ is generated by the elements of the form $\prod\limits_{\lambda} x_{\lambda}^{n_\lambda}$, which are of degree $i$ in $A$.

Ok, I think I'm fine with this proposition. Here's the corollary of its:

Corollary

Let $A = \bigoplus\limits_{i \in \mathbb{Z}} A_i$ be a graded commutative ring with positive degrees (i.e $A_i = 0, \forall i < 0$), $\mathfrak m$ the graded ideal $\bigoplus\limits_{i \ge 1} A_i$. TFAE:

  1. The ideal $\mathfrak m$ is a finitely generated $A-$module.

  2. The ring $A$ is a finitely generated $A_0-$$\color{red}{\mathbf{module}}$.

And here's its proof:

Proof

If a family $(y_\mu)$ of elements of $A$ is a system of generators of the $A-$module $\mathfrak m$ (resp. the $A_0-$module $A$), so is the family consisting of the homogeneous components of the $y_\mu$, and the equivalent of 1., and 2. follows from Proposition 1.

I have no idea how it follows from 1. I also suspect that the word module in red above should be $\mathbf{algebra}$ instead. Since for some strange reasons I cannot prove $1. \implies 2.$

Am I missing something here? :(

Thank you very much,

And have a good day,

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  • 1
    $\begingroup$ Yes, this seems to be a typo. For example, $k[x]$ is a counterexample to the stated claim (the irrelevant ideal is singly generated, but it is not a finite $k$-module). $\endgroup$ – Alex Youcis Feb 17 '14 at 8:40
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    $\begingroup$ @AlexYoucis: Sweet, thanks a lot for your answer. :* I see what you mean. So, basically, if I change the word 'module' in 2. to 'algebra', then it should be okay, right? :) $\endgroup$ – user49685 Feb 17 '14 at 8:44
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    $\begingroup$ Thanks for the kiss. Yeah, that' right. This is exercise 4.5(D) in Vakil, for example. $\endgroup$ – Alex Youcis Feb 17 '14 at 8:53
  • $\begingroup$ The original book (in French) doesn't contain this mistake! $\endgroup$ – user26857 Feb 18 '14 at 16:44
  • $\begingroup$ @user121097: Thank you for pointing this out. Urg, the translation team of Springer pretty sucks. Chapter I, and II are great. Whereas, chapter III, typos, and minor errors appear once every 1, or 2 pages. :( I shudder to think how chapter IV, V, VI, and VII will look like. :( $\endgroup$ – user49685 Feb 19 '14 at 17:52

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