I'm reading the proof of the Artin-Rees lemma in Matsumura's Commutative Ring Theory, and I'm having trouble understanding the proof of the Artin-Rees lemma. The lemma states
Let $A$ be a Noetherian Ring, $M$ a finite $A$-module, $N\subseteq M$ a submodule and $I$ an ideal of $A$. Then there exist a positive integer $c$ such that for every $n>c$ we have $I^nM\cap N=I^{n-c}(I^cM\cap N)$.
Now I'm in doubt of how we get $\eta\in I^{n-c}(I^cM\cap N)$ in the last line. Why is that the case? I get the things stated in the above line, but I Don't see why thay means $\eta\in I^{n-c}(I^cM\cap N)$?
The below questions are clear to me now - thanks!
The proof states that the inclusion $\supset$ is obvious, but I don't see why that is the case?
Secondly it also states that if $I=(a_1,...,a_r)$ and $M=(\omega_1,...,\omega_s)$ then elements of $I^nM$ can be written as $\sum^s_1f_i(a)\omega_s$ where $f_i(a)=f_i(a_1,...,a_r)$ is homogenous of degree $n$ with coefficients in $A$. Why is that the case?
First, $\supset$ is obvious because $I^cM\cap N\subset I^cM$ so $$I^{n-c}(I^cM\cap N)\subset I^{n-c}(I^cM)\subset I^nM.$$ Similarly, $I^cM\cap N\subset N$ so $$I^{n-c}(I^cM\cap N)\subset I^{n-c}N\subset N.$$ Therefore, $I^{n-c}(I^cM\cap N)\subset I^nM\cap N$.
Secondly, elements of $I^n$ are of the form $f(\underline{a})=f(a_1,\ldots, a_r)$ where $f(\underline{x})$ is homogeneous of degree $n$ with coefficients in $A$.
Thirdly, it is impossible to answer this question without knowing where $\nabla$ comes from, what the $d_j$ are, and how $c$ was found.