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I'm currently trying to learn about etale homotopy for schemes as introduced by Artin-Mazur. I know that by the Artin-Mazur comparision theorem, it is possible to compute the etale homotopy type of certain class of varieties as the profinite completion of the complex points. However, in most other cases for schemes, it seems quite cumbersome to calculate the étale homotopy type of a locally noetherian scheme say. Are there any explicit computations of the étale homotopy type that are particularly helpful for understanding the general theory? Or am I missing something here?

Sorry if my question is a bit vague.

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This answer to one of my questions: mathoverflow.net/questions/112007/… has an interesting property, described in the comments, that might be helpful to work out the computations of. Or it might not be - I don't know much about etale homotopy. –  Will Sawin Nov 22 '12 at 2:02

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Here's an example which is, in my opinion, illuminating. It is also quite easy, which I view as a plus.

Namely, consider the étale homotopy type of $\text{Spec}~\mathbb{R}$. By your comparision theorem, this is (pro)-equivalent to $B(\mathbb{Z}/2\mathbb{Z})$. But in fact the (pro)-simplicial set one gets is precisely the bar construction for $G=\text{Gal}(\mathbb{C}/\mathbb{R})=\mathbb{Z}/2\mathbb{Z}$(!!!), showing that computing the étale cohomology of $\text{Spec}~\mathbb{R}$ is "the same" as computing the group cohomology of $G$. This is a good, and not hard, exercise.

In general, if $k$ is a field with finite Galois group $G$, its étale homotopy type will equal(!) the bar construction of $BG$ for $G=\text{Gal}(k^s/k)$; with an appropriate version of $BG$ for $G$ profinite, this will be true for any field.

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Maybe my answer will not fit exactly your question. What I like very much is D. Sullivan's use of Artin-Mazur's theory in his proof of Adams'conjecture. What D. Sullivan does is the computation of the étale homotopy type of the classifying space $BU_n$ of the complex unitary group and he does this computation by considering this classifying space as a direct limit of complex Grassmannians:

$$G_{n,k}\cong GL(n+k,\mathbb{C})/(GL(n,\mathbb{C})\times GL(k,\mathbb{C}))$$

Then he analysises the étale homotopy type of $BU_n$ by looking at its associated arithmetic square. What is important in Sullivan's proof of the Adams'conjecture is the understanding of the action of the absolute galois group on the étale homotopy type of $BU_n$ which has a deep impact on Adams operations in $K$-theory. In his MIT notes "Geometric Topology Localization, Periodicity, and Galois Symmetry" he also states a conjecture, now a theorem: "the Sullivan's conjecture", that has some important implications on the study of the étale homotopy type of real algebraic varieties. Of course all this material can be found in section 5 "Algebraic geometry (étale homotopy type)" of the notes cited above with many examples.

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