As the question title suggests, what is the role cohomology of coherent sheaves plays for SGA 4.5, étale cohomology? Why are they so important for the construction and establishing properties of étale cohomology? What are some of the main essences, intuitions, and theorems of the cohomology of coherent sheaves that are exploited or get "carried over" from the classical setting, in some sense?
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1$\begingroup$ Excuse me, but could you give an example of the usage of coherent sheaves in étale cohomology. I'm a total ignorant, but, as long as I know, only torsion étale sheaves or more generally mixed l-adic sheaves are used at all... $\endgroup$– user40276Commented Oct 7, 2016 at 0:44
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$\begingroup$ @user40276 I'm not quite sure myself, but multiple people have said having intuition wrought out of a good grasp of how stuff works in the classical setting a la cohomology of coherent sheaves carries over well to étale cohomology, and I'm hoping someone can talk about that. Maybe I'm way wrong. $\endgroup$– user97565Commented Oct 7, 2016 at 0:50
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1$\begingroup$ In my novice understanding: I think the usefulness of studying cohomology of coherent sheaves on a scheme is primarily for analogy (although there are bound to be some direct applications of the theory). Coherent sheaves are the class of sheaves where cohomology works 'nicely' in the setting of sheaves of modules on a (separated and Noetherian, say) scheme. There are analogous classes of sheaves ($\ell$-adic constructible sheaves, etc.) for the étale cohomology which are the 'nice' ones in that setting. Many similar formal properties hold, and the derived functor formalism unites the two. $\endgroup$– dorebellCommented Oct 7, 2016 at 2:45
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$\begingroup$ Are you asking about coherent sheaves for the etale topology? One of the first thing one shows (using faithfully flat descent) is that, essentially, $\mathrm{QCoh}(X_{\acute{e}\text{t}})\cong \mathrm{QCoh}(X_{\mathrm{Zar}}$ in the obvious way (namely if $\mathcal{F}$ is quasi-coherent on $X_{\acute{e}\text{t}}$ then its pullback to the Zariski site is a quasi-coherent) and that this preserves global sections. In particular, the etale cohomology is the same as the Zariski cohomology of coherent sheaves. In other words NOTHING new is gained by thinking about quasi-coherent sheaves $\endgroup$– Alex YoucisCommented Oct 7, 2016 at 5:57
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4$\begingroup$ Since this question has gained 13 upvotes in 13 hours, I assume that many people are here who understand and appreciate it, but actually I don't understand what is being asked here (although I am familiar with the concepts of cohomology and coherent sheaves). Could someone explain this to me, in his own words? If I take the question literally, I don't know what to say except for that both notions of cohomology are actually the same, just applied to different sites. $\endgroup$– HeinrichDCommented Oct 7, 2016 at 6:59
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1 Answer
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You ask two different questions, I believe.
1) When Grothendieck invented étale cohomology, cohomology of coherent sheaves was already well known, thanks to Oka/Cartan's theorems A and B, Serre's FAC, Gaga, and the Serre-Grothendieck duality theorems). It was also well known (Weil, already) that the invention of some cohomology theory closer to singular cohomology was one path towards the Weil conjecture. In some sense, cohomology of coherent sheaves was an insufficient prototype.
2) Coherent sheaves naturally furnish examples of étale sheaves and Grothendieck's descent theorems imply that nothing is gained by viewing them in the étale framework. This implies in particular that étale line bundles or étale vector bundles are the same objects than line bundles or vector bundles. This gives at least some étale sheaves, such as $(\mathbf G_m)_X$ or $\mathrm{GL}(n)_X$, whose cohomology can be computed; for example: $$ H^1_{\rm ét}(X,\mathbf G_m)=H^1_{\rm Zar}(X,\mathbf G_m)=\mathop{\rm Pic}(X).$$
For $n$ invertible on $X$, the Kummer exact sequence $$ 1\to (\mu_n)_X \to (\mathbf G_m)_X\xrightarrow{x\mapsto x^n}(\mathbf G_m)_X\to 1 $$ gives rise to a long exact sequence which is the starting path to the computation of étale cohomology groups with coefficients in $\mu_n$.
When $X$ is an $\mathbf F_p$-scheme, the Artin-Schreier exact sequence $$ 0 \to (\mathbf Z/p\mathbf Z)_X \to (\mathbf G_a)_X \xrightarrow{x\mapsto x^p-x} (\mathbf G_a)_X\to 0 $$ allows to compute the étale cohomology groups of $X$ with coefficients in $\mathbf Z/p\mathbf Z$. One sees in particular that they vanish beyond the dimension of $X$, hence are insufficient for the Weil conjectures.
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2$\begingroup$ As it stands, the formulation is awkward since G_m-valued cohomology is presented as an instance of coherent cohomology. $\endgroup$ Commented Oct 7, 2016 at 18:39
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$\begingroup$ Thanks @MatthieuRomagny. What I had in mind, as you probably understood, is that the first étale/Zariski cohomology group of $\mathbf G_m$ classifies étale/Zariski line bundles, and the latter are quasi-coherent sheaves. $\endgroup$– ACLCommented Oct 7, 2016 at 22:02
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$\begingroup$ Dear ACL, sure, the comment is apt. $\endgroup$ Commented Oct 8, 2016 at 10:19