I'm studying Eisenbud's Commutative Algebra Book, but I need help with the topic of Filtrations and the Artin- Rees Lemma. I'm stuck with the proposition 5.2, I don't even get the first '"clearly" in the proof, could anyone detail this and add the intermidiate steps of the rest of the proof please?
Proposition. Let $I$ be an ideal in a ring $R$, and suppose that $M$ is a finitely generated $R$-module. If $J : M = M_0 \supseteq M_1 \supseteq \ldots$ is an $I$-stable filtration by finitely generated submodules of $M$, then $\text{gr}_J M$ is a finitely generated module over $\text{gr}_I R$.
Proof. Suppose that $IM_i = M_{i+1}$ for all $i \geq n$. Clearly, $(I/I^2)(M_i/M_{i+1}) = M_{i+1}/M_{i+2}$ for $i \geq n$. Thus the union of any sets of generators of the modules $M_0/M_1,\ldots, M_n/M_{n+1}$ will generate $\text{gr}_J M$. Since each $M_i$ is finitely generated, each of these sets of generators may be chosen to be finite.
Many thanks.
The calculation takes place inside of $\mathrm{gr}_J(M) = \bigoplus_{i \geq 0} M_i/ M_{i+1}$ with the $\mathrm{gr}_I(R)$-module structure induced by the $\mathbb{Z}$-linear maps $$\odot_{k,i} : I^k/I^{k+1} \otimes_{\mathbb{Z}} M_i/M_{i+1} \to M_{i+k}/M_{i+k+1}, \quad [u] \otimes [m] \mapsto [u \cdot m].$$ Let us denote that by $\odot_{k,i}$ just to avoid any confusion with different module structures. I think your confusion is caused by the usual abuse of notation to just omit every scalar multiplication and pretend they are equal.
Consider $k = 1$. Then, the image of $$\odot_{1,i} : I/I^2 \otimes_{\mathbb{Z}} M_i/M_{i+1} \to M_{i+1}/M_{i+2}$$ is equal to $(I \cdot M_i)/(M_{i+2})$, and this is $M_{i+1} / M_{i+2}$ for $i \geq n$. Thus, $\odot_{1,i}$ is surjective for $i \geq n$. This is meant with the equation "$I/I^2 \cdot M_i/M_{i+1} = M_{i+1}/M_{i+2}$".
The $R$-modules $M_0,\dotsc,M_n$ are finitely generated. Choose finite generating sets $E_0,\dotsc,E_n$. It is claimed that their images in $M_0/M_1,\dotsc,M_n/M_{n+1}$ provide a generating set of $\mathrm{gr}_J(M)$ over $\mathrm{gr}_I(R)$.
But we know that $\odot_{1,n}$ is surjective. It follows that every element in $M_{n+1}/M_{n+2}$ is a $I/I^2$-linear combination of elements in $M_n/M_{n+1}$ (inside of $\mathrm{gr}_J(M)$). But this means that the generators of $M_n/M_{n+1}$ are sufficient to generate $M_{n+1}/M_{n+2}$. And so on. We get all of $M_i / M_{i+1}$ for $i \geq n$.