I'm trying to understand lemma 10.23 from Atiyah and Macdonald.
Lemma 10.23. Let $\phi:A\rightarrow B$ be a homomorphism of filtered groups, i.e. $\phi(A_{n})\subset B_{n}$, and let $G(\phi):G(A)\rightarrow G(B)$, $\hat{\phi}:\hat{A}\rightarrow\hat{B}$ be the induced homomorphisms of the associated graded and completed groups. Then
i) $G(\phi)$ injective $\implies$ $\hat{\phi}$ injective;
ii) $G(\phi)$ surjective $\implies$ $\hat{\phi}$ surjective.
My Question: What is exactly meant with $G(\phi)$ and $\hat{\phi}$? As mentioned in the comment, my guess was correct.
My guess would be:
Notice that $G(A) = \oplus_{n\geqslant 0} A_{n}/A_{n+1}$. So I think that $G(\phi)$ consists of the maps $G_{n}(\phi):A_{n}/A_{n+1}\rightarrow B_{n}/B_{n+1}, a_{n} + A_{n+1}\mapsto \phi(a_{n})+B_{n+1}$. Which is clearly well-defined since $\phi(A_{n+1})\subset B_{n+1}$.
And for $\hat{\phi}$ I guess that it is defined by $\alpha_{n}:A/A_{n+1}\rightarrow B/B_{n+1}, a+A_{n+1}\mapsto \phi(a)+B_{n+1}$.
Is this the right way of thinking about these maps?
New Question: I also have a question about the proof of this lemma. To give all the information needed here is a link of the book from Atiyah and MacDonald http://www.math.toronto.edu/jcarlson/A--M.pdf since this question already becomes pretty long. I have problems with the final conclusion. Where they say that by taking the inverse limit and applying lemma 10.2 gives the desired result. For instance, in lemma 10.2 we have three inverse systems, while here we have only two. So what is the exact sequence one uses to apply lemma 10.2 on?