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:
The ideal of $A$ generated by the family $(x_\lambda)_{\lambda \in L}$ is equal to $\mathfrak{m}$
The family $(x_\lambda)_{\lambda \in L}$ is a system of genereators of the $A_0-$algebra $A$.
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:
The ideal $\mathfrak m$ is a finitely generated $A-$module.
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,