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I am reading the book Introduction to commutative algebra by Atiyah and Macdonald. I have some questions about Corollary 10.3 and Corollary 10.4.

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Why the sequence $$ 0 \to \frac{G'}{G' \cap G_n} \to \frac{G}{G_n} \to \frac{G''}{pG_n} \to 0 $$ is exact? The map $g: \frac{G}{G_n} \to \frac{G''}{pG_n}$ is given by $a + G_n \mapsto pa + pG_n$? Why $\ker g = G'/(G' \cap G_n)$?

Why $\hat{G'} = \varprojlim G'/(G' \cap G_n)$ but not $\hat{G'} = \varprojlim G'/G'_n$?

In the third and fourth lines of Page 105, if $G'=G_n$, then $G'' = G/G' = G/G_n$. But why $\hat{G''}=G''$?

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Your first question: Lets call the first arrow $f:G' \rightarrow G$. The kernel of the morphism $G/G_n \rightarrow G''/pG_n \rightarrow 0$ is $\frac{ker(p)+G_n}{G_n}$ which is equal to $\frac{im(f)+G_n}{G_n}$ which is isomorphic to $G'/G'\cap G_n$ (you can directly establish this isomorphism).

Your second question: $G_n'=G' \cap G_n$.

Your third question (+1): One way to see that $\hat{G''}=G''$ is to find what is the fundamental system of neighborhoods of zero in $G/G_n$ that is induced by the fundamental system of neighborhoods of zero in $G$. We can see that this is just the finite set of cosets $\frac{G_i+G_n}{G_n}$ for $i=0,1,\cdots,n$. Next, lets construct the inverse limit induced by this topology on $G/G_n$. Since the sequence of open sets is finite, ending at $0$, we see that every coherent sequence in $G/G_n$ will have length $n+1$ and its last element is just an element of $G/G_n$. This observation can be used to establish an isomorphism $G/G_n \rightarrow \hat{G/G_n}$ by taking every element of $G/G_n$ to the coherent sequence induced by that element.

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