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One of the most standard way to construct Galois representations is the geometric way: one starts from a variety $X$ defined over $\bf Q$ say; the Galois group acts on ${\overline X}:= X \times {\rm Spec}(\overline {\bf Q})$, "hence" on the etale cohomology groups $H^n_{et}({\overline X}, {\bf Q}_\ell)$: the construction is given in many texts and online articles.

I am looking for a reference explaining the CONTINUITY of the above $\ell-$adic representations (which is done by hand for alliptic curves, but I am looking for a proof in generality of this continuity) -- this is certainly a consequence of the definition of the étale cohomology (taking projective limits), but I couldn't find a proof of it...

Thanks!

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  • $\begingroup$ I think you would also need to know something about finiteness of cohomology groups. $\endgroup$
    – Zhen Lin
    Jul 18, 2015 at 2:25
  • $\begingroup$ Of course! sorry, I forgot to add the standard hypotheses on $X$ insuring that the $H^n$ are finite dimensional... $\endgroup$
    – dionysos
    Jul 18, 2015 at 21:02

2 Answers 2

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Here is a nice conceptual way which makes the continuity much clearer. It may not be in the form that you want, but you can certainly use it to prove the result in your language.

So, let's assume that $f:X\to\mathrm{Spec}(\mathbb{Q})$ is smooth proper (as we always do). We then have the following nice fact:

Theorem(étale Ehrassman's theorem): Let $f:X\to Y$ be smooth proper. For any LCC sheaf $\mathcal{F}$ on $X$, the sheaves $R^if_\ast\mathcal{F}$ are LCC on $Y$.

In particular, note that for our $f:X\to\mathrm{Spec}(\mathbb{Q})$ we have that $R^if_\ast\underline{\mathbb{Z}/\ell^n\mathbb{Z}}$ is an LCC sheaf on $\mathrm{Spec}(\mathbb{Q})$. But, by standard theory, this is equivalent to giving a continuous, finite $G_\mathbb{Q}$-module $M_n$. In particular, since the obvious compatibilities hold, one sees that

$$M:=\varprojlim M_n=\varprojlim R^if_\ast\underline{\mathbb{Z}/\ell^n\mathbb{Z}}$$

is a continuous $\mathbb{Z}_\ell$-representation of $G_\mathbb{Q}$.

Why does this matter to us? Well, by smooth proper base change, if $\overline{x}:\mathrm{Spec}(\overline{\mathbb{Q}})\to\mathrm{Spec}(\mathbb{Q})$ is any geometric point, then

$$(R^if_\ast\underline{\mathbb{Z}})_\overline{x}=H^i_\mathrm{\acute{e}t}(X_{\overline{\mathbb{Q}}},\mathbb{Z}/\ell^n\mathbb{Z})$$

But, taking stalks is precisely the equivalence

$$\left\{\begin{matrix}\text{LCC sheaves}\\\text{on }\mathrm{Spec}(\mathbb{Q}\end{matrix}\right\}\longleftrightarrow\left\{\begin{matrix}\text{finite continuous}\\G_\mathbb{Q}\text{-modules}\end{matrix}\right\}$$

Thus, you see that $M_n=H^i_{\mathrm{\acute{e}t}}(X_\overline{\mathbb{Q}},\mathbb{Z}/\ell^n\mathbb{Z})$ as $G_\mathbb{Q}$-modules, from where continuity follows. And, of course, $M=H^i_{\mathrm{\acute{e}t}}(X_{\overline{\mathbb{Q}}},\mathbb{Z}_\ell)$ as $G_\mathbb{Q}$-modules.

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  • $\begingroup$ Thanks a lot Alex. I am happy to know that a proof exists -- though, as you write, I'd like to find one as "down to earth" as possible. So many thanks (for non-expert people like me, this is very helpful). I was about to post this: Conrad-notes: see 1.3.3 p44, I think that this is very close to your explanation. Conard mentions that the continuity can be checked directly -- maybe he was thinking of your proof? $\endgroup$
    – dionysos
    Jul 21, 2015 at 13:42
  • $\begingroup$ @dionysos No problem! Glad I could help. I'll have to think about the easiest way to state continuity without thinking this way. You see, the problem is, as you pointed out, it's enough to show continuity of the action on $H^i(\overline{X},\mathbb{Z}/\ell^n\mathbb{Z})$. This amounts to saying that $G_\mathbb{Q}\to \mathrm{GL}_{\mathbb{Z}/\ell^n\mathbb{Z}}(H^i(\overline{X},\mathbb{Z}/\ell^n \mathbb{Z}))$ factors through a finite quotient. This quotient is not obvious to me except to say that $R^if_\ast\underline{\mathbb{Z}/\ell^n\mathbb{Z}}$ is of the form $\endgroup$
    – Alex Youcis
    Jul 21, 2015 at 13:51
  • $\begingroup$ $\mathrm{Hom}_{\mathrm{Spec}(\mathbb{Q})}(-,\mathrm{Spec}(L))$ for some number field $L$. The kernel is then $\mathrm{Gal}(\overline{\mathbb{Q}}/L)$. I'm not sure how to state this intrinsically in this generality. For elliptic curves it's simple since $H^1$ is dual to $T_\ell$ which clearly is continuous, and the higher $H^i$ are just wedges. I don't know something as concrete as 'adjoin torsion points to $\mathbb{Q}$', like for ab. vars, in the generality you ask. Let me know if you think of any neat way of characterizing $L$. $\endgroup$
    – Alex Youcis
    Jul 21, 2015 at 13:53
  • $\begingroup$ I tried this way, but I am not sure if the Galois action necessarily factors thru a finite quotient (it is true for complex representations, but I [think I] saw it is not always true in the l-adic case) ? $\endgroup$
    – dionysos
    Jul 21, 2015 at 15:21
  • $\begingroup$ @dionysos It is for the finite pieces--the Z/l^nZ coeffs. $\endgroup$
    – Alex Youcis
    Jul 21, 2015 at 15:26
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Check the following materials: http://web.mit.edu/~corwind/www/jp11.pdf and https://www.math.leidenuniv.nl/scripties/KretMaster.pdf.

Enjoy!

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  • $\begingroup$ Thanks for this nice paper which conveniently explains a good deal of the "well known" properties.. I recommend it as well.. But ths continuity is treated in the cas in an elliptic curve, but I don't see it for other etale cohomology groups? $\endgroup$
    – dionysos
    Jul 17, 2015 at 14:20

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