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Let $K$ be a number field, $R$ the ring of integers of $K$, ${\mathbf{A}^f}$ the ring finite adeles of $K$, and ${\widehat{R}}\subset {\mathbf{A}^f}$ the ring of integral adeles.

Let $G$ be an affine group scheme of finite type over $R$ with smooth generic fiber $G_K$. Let $$c(G)=G({\widehat{R}})\backslash G({\mathbf{A}^f})/G(K)$$ denote the pointed set of double cosets of $G$. If I understand correctly, Yevsey Nisnevich in his Comptes rendus note of 1984 claims that there exists an exact sequence of pointed sets $$ 1\to c(G)\to H^1_{\rm et}({\rm Spec}(R),G)\to H^1(K,G)\times \prod_{{\mathfrak{p}}\in{\rm Spec}(R)\smallsetminus \{0\} } H^1_{\rm et}({\rm Spec}(R_{\mathfrak{p}}),G), $$ where $R_{\mathfrak{p}}$ denotes the completion of $R$ at the finite place ${\mathfrak{p}}$ of $K$.

Question. How can one define the map $c(G)\to H^1_{\rm et}({\rm Spec}(R),G)$ in this exact sequence?

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  • $\begingroup$ In my paper with Yuri Tschinkel, Torseurs arithmétiques et espaces fibrés, we have a similar description (Proposition 1.2.6). NB. As remarked by Philippe Gille, there is a slight mistake there, that we consider torsors which are locally trivial for the Zariski topology, without saying so. $\endgroup$
    – ACL
    May 14, 2015 at 8:33
  • $\begingroup$ @ACL: Just a comment; as remarked in my answer, if $G$ satisfies weak approximation, the image of the map in question is precisely $H^1_{Zar}(\text{Spec}(R), G)$. $\endgroup$ May 14, 2015 at 17:56

1 Answer 1

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Given an element of $c(G)$, we are looking for an etale $G$-torsor which is trivial when restricted to $\text{Spec}(K)$ and each $\text{Spec}(R_\mathfrak{p})$, for $\mathfrak{p}$ a finite prime of $R$. Because this $G$-torsor will be trivial over $K$, it will in fact be trivial over some Zariski-open $U$ of $\text{Spec}(R)$, say $U=\text{Spec}(R[1/N])$. Furthermore, for each $\mathfrak{p}$ dividing $N$, this torsor will be trivial on $\text{Spec}(R_\mathfrak{p})$.

Summing up (and using fpqc descent), we are looking for a trivial $G$-torsor on $$U':=U\cup \bigcup_{\mathfrak{p}|N}\text{Spec}(R_\mathfrak{p})$$ and descent data to $\text{Spec}(R)$. I claim one can extract such descent data from an element of $c(G)$.

Namely, let's choose a trivialization of our trivial $G$-torsor on $U'$. Now the descent data boils down to choosing elements of $G(\text{Frac}(K_\mathfrak{p}))$ for each $\mathfrak{p}|N$. Since descent is effective for the cover $U'\to\text{Spec}(R)$, we may harmlessly enlarge $U'$ to $$U''=U\cup \bigcup_{\mathfrak{p}\in R}\text{Spec}(R_\mathfrak{p}).$$ Now descent data is given by a choice of element of $G(R_\mathfrak{p})$ for each $\mathfrak{p}\in U$ and an element of $G(K_\mathfrak{p})$ for each $\mathfrak{p}|N$. That is, descent data (for this cover, after choosing a trivialization!) is an element of $G(\mathbf{A}^f)$. But changing the trivialization over $U$ (possibly after shrinking $U$) changes this descent data by an element of $G(K)$; changing the trivialization on $\bigcup_{\mathfrak{p}\in R}\text{Spec}(R_\mathfrak{p})$ changes the descent data by an element of $G(\hat R)$. Thus the desired double coset space precisely corresponds to fpqc descent data.

Let me be a bit more explicit, since there's been a question in the comments. Let $$V=\text{Spec}(K)\cup \bigcup_{\mathfrak{p}\in R} R_\mathfrak{p}=\varprojlim_U U\cup \bigcup_{\mathfrak{p}\in R}\text{Spec}(R_\mathfrak{p}).$$ The above discussion shows that (1) descent is effective for the cover $V\to \text{Spec}(R)$ and (2) elements of $c(G)$ are in bijection with descent data on a trivialized $G_V$ torsor over $V$, modulo choice of trivialization. Hence one may associate to an element of $c(G)$ descent data; as descent is effective this gives an actual $G$-torsor on $\text{Spec}(R)$ as desired.

Strictly speaking, we've given a map from the desired double coset space to $H^1_{\text{fpqc}}(\text{Spec}(R), G)$. We need to argue that in fact the torsor we get is etale-locally trivial. In fact the torsor we've given is clearly trivial over some open set, by construction (namely, choosing a representative of the double coset, there is an open set $U$ consisting of $\mathfrak{p}$ so that the relevant element of $G(K_\mathfrak{p})$ is in fact in $G(R_\mathfrak{p})$). So we must argue that in fact our torsor is etale-locally trivial in a neighborhood of the other $\mathfrak{p}$. I don't see why this is true if $G$ doesn't satisfy weak approximation; if it does, we can always make our descent data integral at any given $\mathfrak{p}$ by modifying it by an element of $G(K)$, and then conclude by the previous sentence. But perhaps I'm missing something easy. (See grghxy's comment for the observation that weak approximation is not necessary, b/c of Artin approximation. If weak approximation is satisfied, the map in fact lands in $H^1_{Zar}(\text{Spec}(R), G)$.)

For more details on this sort of construction, see e.g. this answer. I'm not an expert on this sort of thing, so caveat emptor.

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  • $\begingroup$ It isn't true that $G_K$-torsors must be trivial; that only holds under certain hypotheses on $G_K$ (and $K$). $\endgroup$
    – grghxy
    May 14, 2015 at 7:30
  • $\begingroup$ @grghxy: We're looking for a trivial $G_K$-torsor, since we're mapping into the kernel of a map to $H^1(K, G_K)\times \cdots$ $\endgroup$ May 14, 2015 at 7:32
  • $\begingroup$ (Of course I agree that not all $G_K$-torsors will be trivial, e.g. if $G$ is $PGL_n$.) $\endgroup$ May 14, 2015 at 7:35
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    $\begingroup$ Very true, but I had misread your answer to think you were saying something else. Now all is clear; doesn't the link you give to a related question also answer this one (assuming $G$ is flat and affine of finite type over the Dedekind base)? $\endgroup$
    – grghxy
    May 14, 2015 at 7:38
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    $\begingroup$ @DanielLitt Could you please describe the construction more clearly? We start from an element $g\in G(\mathbf{A}^f)$. How do we construct the corresponding torsor? From your exposition is seems that you start from a torsor.... $\endgroup$ May 14, 2015 at 14:21

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