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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:… 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

3 Answers 3

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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This is kind late, but here are some examples to add on which might demonstrate some of the elementary tools that one has in this subject:

  • As Daniel said, if $G = Gal(k_{sep}/k)$ then $BG \simeq Spec\,k_{et}$. Perhaps the easiest way to see this is the observation that if $L/k$ is a Galois extension then $L \otimes_k L \simeq \prod_{\sigma \in Gal(L/k)} Spec\,L$ so that, upon applying $\pi_0$ we get the simplicial set $BGal(L/k)$.

  • We can do $\mathbb{G}_m$ for $k$ being separably closed explicitly (without comparison theorems). With any scheme in which cohomology may be computed using Cech theory, the etale homotopy type may be computed using 0-coskeletal hypercovers, i.e. Cech covers in the homotopy category of hypercovers. We have a left final subcategory of Cech covers of the form $cosk_0(G_m) \rightarrow G_m$ induced by $G_m \rightarrow G_m, t \mapsto t^n$. This can be proved using, say, Riemann Hurwitz arguments (if you want to avoid comparison theorems completely). Observing that $cosk_0(G_m)_n \simeq G_m^{n+1} \simeq G_m \times \mu_n^n$, $\pi_0$ again gets us $B\mu_n$ and the etale homotopy type is the pro-space $\{B\mu_n\}$ indexed by divisibility.

  • Of course there are comparison theorems which might be "cheating" but here's one that is interesting. If you're a projective variety which has a model over $W(\bar{F}_p)$, then the etale homotopy type over $\mathbb{C}$ and over $\bar{F}_p$ is equivalent - this is a direct consequence of proper and smooth base change so that might tell you that the theory encodes difficult theorems in algebraic geometry.

  • In Friedlander's thesis (his completion of Quillens sketch of the complex Adams conjecture using etale homotopy theory), he computed the homotopy fiber of $\mathcal{E} - 0(X) \rightarrow X$ where $\mathcal{E}$ is a vector bundle of rank $r \geq 2$ and $0(X)$ is the zero section. He proved that this is (upon completion away from the residue characteristics), the $2r-1$ sphere. Roughly speaking, Friedlander does two things - he showed that for $x \rightarrow X$ a geometric point, $\mathcal{E}- 0(X)_x$ is indeed equivalent to the appropriately completed sphere. This step uses the general "six functors" type calculations for smooth pairs which can be found in Milne VI.5, say. The other step compares the fiber versus the homotopy fiber upon take etale homotopy types. This uses a comparison between the Serre spectral sequence (of fibrations between (pro)-spaces) and the Leray spectral sequence (as composition of functors).

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