How is etale cohomology of integer rings related to Galois cohomology? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T09:18:38Z http://mathoverflow.net/feeds/question/62607 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/62607/how-is-etale-cohomology-of-integer-rings-related-to-galois-cohomology How is etale cohomology of integer rings related to Galois cohomology? David Loeffler 2011-04-22T09:29:30Z 2011-04-23T07:28:39Z <p>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</p> <p><code>$H^i_{\mathrm{et}}(\operatorname{Spec} O_{K, S}, M),$</code></p> <p>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 $S$.</p> <p>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?</p> http://mathoverflow.net/questions/62607/how-is-etale-cohomology-of-integer-rings-related-to-galois-cohomology/62658#62658 Answer by Olivier for How is etale cohomology of integer rings related to Galois cohomology? Olivier 2011-04-22T19:54:34Z 2011-04-23T07:28:39Z <p>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. </p> <p>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.</p> <p>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.</p> <p>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.</p> <p>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.</p>