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In the paper of Bloch and Kato in the Grothendieck Festschrift, and some other papers relating to the Bloch-Kato conjecture and the ETNC, the cohomology groups

$H^i_{\mathrm{et}}(\operatorname{Spec} O_{K, S}, M),$

seem to come up often, where $K$ is a number field, $S$ is a finite set of places of $K$, and $M$ is a finite or profinite $G_K = \operatorname{Gal}(\overline{K} / K)$-module unramified at primes outside $S$.

How should one think about these cohomology groups? How are they related to the much more familiar (to me at least) continuous Galois cohomology groups $H^i(G_K, M)$ (or the restricted ramification analogues $H^i(G_{K, S}, M)$)? Why are they the more natural things to work with in this context?

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I asked the same question here: mathoverflow.net/questions/60310/… –  Timo Keller Apr 22 '11 at 10:57
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One thing is that Galois cohomology, by its very definition, requires the choice of a base-point (the `biggest' extension in which the primes outside $S$ are unramified), while etale cohomology is defined intrinsically in terms of the etale site over $\text{Spec}\mathcal{O}_{K,S}$. Usually, when you actually compute things, you end up returning to group cohomology, but it seems nicer to set up the theory without it. –  Keerthi Madapusi Pera Apr 22 '11 at 13:44
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There is quite a bit on this in the second chapter of Milne's Arithmetic Duality Theorems --- see for example Proposition 2.9, the discussion on pages 195--197 (of the second edition), and Lemma 5.5. –  mephisto Apr 22 '11 at 14:46
    
There's an old paper of Mazur that discusses these kinds of groups, though it's been a long time since I looked at it so I don't recall what the specific goals of the paper are. I think it's called "Notes on the etale cohomology of number fields." It might be worth a look. –  Ramsey Apr 22 '11 at 16:29
    
@Keerthi: I think that Serre in his book on Galois cohomology talks about how one can do Galois cohomology "intrinsically"---probably all that it boils down to though is that he's doing etale cohomology really :-). Serre defines $H^i(k,M)$ without making a specific choice of alg closure of $k$. –  Kevin Buzzard Apr 22 '11 at 21:32
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up vote 9 down vote accepted

By usual (sometimes not so trivial) homological arguments, one can reduce to the case where $M$ is a finite discrete module over an artinian ring of residual characteristic $p$. In that case, I think you want $S$ to contain places above $p$ as well, even if your $M$ is unramified at $p$, so let me assume this.

The module $M$ induces an étale sheaf $M_{et}$ on $\operatorname{Spec}\mathcal O_{L,S}$ for all finite extension $L/K$. The spectral sequence UPDATE (converging to $H^{i+j}(\operatorname{Spec}\mathcal O_{K,S},M_{et})$) $$E_{2}^{i,j}=\underset{\longrightarrow}{\operatorname{\lim}}\ H^{i}(\operatorname{Gal}(L/K),H^{j}(\operatorname{Spec}\mathcal O_{L,S},M_{et}))$$ then induces isomorphisms between $E_{2}^{i,0}$ and $H^{i}(\operatorname{Spec}\mathcal O_{L,S},M_{et})$ or in other words $H^{i}(G_{K,S},M)$ is isomorphic to $H^{i}(\operatorname{Spec}\mathcal O_{K,S},M_{et})$. So you can assume that you are working with Galois cohomology throughout $provided$ you use Galois cohomology with restricted ramification.

Because the Tamagawa Number Conjectures are formulated only in the setting above, Bloch and Kato could have used Galois cohomology instead of étale cohomology everywhere without changing anything. To touch upon your last question, I think there are two reasons why they chose étale cohomology.

First, at least at the time they wrote, Galois cohomology was not the most familiar object of the two. In fact, many classical well-known results were given correct complete proofs only very late (in the late 90s in some cases). On the other hand, SGA (and works of Bloch and Kato themselves) existed as references for étale cohomology.

Second, using étale cohmology, one can formulate the TNC over more general bases than $\operatorname{Spec}\mathcal O_{K,S}$ (for instance any scheme of finite type of $\mathbb Z[1/p]$). This kind of generalization had been the key idea of previous works of Kato and Bloch-Kato on higher class field theory so it is not surprising that they decided to at least allow the same kind of generality in their subsequent works.

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"In fact, many classical well-known results were given correct complete proofs only very late (in the late 90s in some cases)." Which results are you thinking of? –  mephisto Apr 22 '11 at 21:49
    
This is probably just me being slow, but could you elaborate a bit more on the spectral sequence argument? (For instance, what is this spectral sequence converging to?) –  David Loeffler Apr 23 '11 at 7:11
    
@Mephisto Central to the approach of Kato is the fact that the étale cohomology of a perfect complex of étale sheaves is a perfect complex. In the early 90s, I wouldn't know what source to quote for the corresponding statement in Galois cohomology. But even something as "basic" as Poitou-Tate had no clear reference before the first edition of Milne's ADT, and that's only from 1986. –  Olivier Apr 23 '11 at 7:32
    
Or think of the excision map in Galois cohomology with restricted ramification. –  Olivier Apr 23 '11 at 8:30
    
@Loeffler. The spectral sequence and a proof of its degeneration can be found in Milne's ADT II Proposition 2.9. @Olivier. The results you mention are hardly "classical Well-known results". As far as I know, all such results had correct complete proofs in the literature by the mid-80s or earlier (not late 90s). For example, Poitou-Tate is proved in Haberland 1978. By contrast, the first complete proof in the literature of the very basic Artin-Verdier duality theorem is in ADT (1986), but that contained an error which wasn't fixed until the second edition (2004). –  mephisto Apr 23 '11 at 11:17
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$\DeclareMathOperator{\h}{H}$ I know this is a bit late, but here is a low-brow answer. Let $X$ be a connected scheme such that any finite subset of $X$ is contained in an open affine subscheme. In this case, we have the following

Theorem. For any sheaf of abelian groups $\mathscr{F}$ on $X$, the natural map $\operatorname{\check H}^\bullet(X,\mathscr{F})\to \h^\bullet(X,\mathscr{F})$ is an isomorphism.

For a proof see M. Artin's paper On the joins of Hensel rings.

Now let $x\to X$ be a geometric point. For an abelian sheaf $\mathscr{F}$, the stalk $\mathscr{F}_x$ is a $\pi_1(X,x)$-module. I claim that $\h^\bullet(X,\mathscr{F})=\h^\bullet\left(\pi_1(X,x),\mathscr{F}_x\right)$. To see this, note that $\h^\bullet(\pi_1(X,x),\mathscr{F}_x)$ is the cohomology of $\mathscr{F}_x$ considered as a sheaf on $\pi_1(X)\text{-}\mathsf{set}$, the category of finite $\pi_1(X)$-sets. Any etale cover $\{U_\alpha\to X\}$ is majorized by a cover $U\twoheadrightarrow X$, and the Čech complex of $\mathscr{F}$ relative to the covering $\{U\to X\}$ is exactly the Čech complex of $\mathscr{F}_x$ relative to the covering $\operatorname{Fib}_x(U)\to \ast$ in $\pi_1(X)\text{-}\mathsf{set}$. In other words, $\h^\bullet(X,\mathscr{F})$ and $\h^\bullet(\pi_1(X),\mathscr{F}_x)$ are computed using the same complexes, so they are naturally isomorphic.

Since $G_{k,S} = \pi_1(\operatorname{Spec} \mathcal{O}_{k,S})$, this recovers your question.

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I'm not sure your last line is true. We have $\pi_1(\mathbb{Z})=0$ (Minkowski) which is not the same as $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$. –  Daniel Loughran Sep 27 '13 at 19:21
    
I don't see how that contradicts Daniel M.'s statement. –  Keerthi Madapusi Pera Sep 27 '13 at 19:31
    
@KeerthiMadapusiPera is quite right. The group $G_{\mathbb{Q}}$ is not a restricted ramification group $G_{\mathbb{Q},S}$ for any finite set $S$ of primes. –  Daniel Miller Sep 27 '13 at 20:43
    
Right sorry I misinterpreted your notation. –  Daniel Loughran Sep 27 '13 at 21:00
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