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I posted my question here, but there is no reply yet. So, I guess I should post it on mathoverflow.

I am reading the book of Schneider about Galois representation and $(\varphi, \Gamma)$-module, Section 1.7, but I don't understand his proof on Lemma 1.7.6. In this section, he introduced the weak topology on the two dimensional local field $\mathscr{A}_L$, where $L$ is a finite extension of $\mathbb{Q}_p$, with $\pi$ is a uniformizer of L. $\mathscr{A}_L$ can be identified with the ring of infinite Laurent series, with coefficients go to $0$ when the indexes go to $-\infty$. Let $k_L$ be the residue field of $L$, then the field $k_L((X))$ is the residue field of $\mathscr{A}_L$.

As I understand, we want to equip the topology on $\mathscr{A}_L$ such that the projection map from $\mathscr{A}_L$ to $k_L((X))$ is continuous, so we can choose a system of neighborhoods near $0$ (on $\mathscr{A}_L$) defined by

$$U_m := X^m O_L[[X]] + \pi^m \mathscr{A}_L$$

Next, in Lemma 1.7.6, he proved that with this topology, $\mathscr{A}_L$ is Hausdorff and complete, but I don't understand the proof. After checking the inverse limit identity, he says "it remains to prove the multiplication map is continuous". Why does it imply the statements? Could anyone help?

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    $\begingroup$ I can't access the book from here, but what might be going on is that the inverse limit proves the claimed result on some open subset (e.g. the ring of integers), and then the multiplication is used to spread this out to the entire field. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Dec 28, 2017 at 18:50
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    $\begingroup$ I also do not have access to your reference, but in the mixed characteristic case that you are considering a choice of local parameters for ${\mathcal A}_L$, say $t_1, t_2=\pi$ determines a canonical lifting $h:L\rightarrow {\mathcal O}_{{\mathcal A}_L}$ which then is used to lift a base of open neighbouhoods of $0$ in $L$ to a base of open neighbouhoods of $0$ in ${\mathcal A}_L$. The difficult part is the definition of the lifting, and some details are in maths.nottingham.ac.uk/personal/ibf/vlm/partI.pdf in pages 9 and ff. Continues in the next remark $\endgroup$
    – F Zaldivar
    Commented Dec 28, 2017 at 20:29
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    $\begingroup$ Continuation: In the same reference there are some details for the proof of the (sequentially) continuity of the multiplication map, which is needed to complete the proof that ${\mathcal A}_L$ is indeed a topological field. $\endgroup$
    – F Zaldivar
    Commented Dec 28, 2017 at 20:32

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