I was reading Quillen's paper on "Some remarks on etale homotopy theory and a conjecture of Adams". Up to some fact about etale version of spherical fibrations which was later proved by Friedlander, Quillen gives a proof of Adams conjecture using etale homotopy theory. Somewhere in middle of paper he is saying: Assume $Q$ is a vector bundle over variety $X$ in characteristic $p$. Assume that $Q^{(p)}$ is the pullback of $Q$ over $X$ by Frobenius map $\Phi : X\to X$ then there is a morphism $ F: Q \rightarrow Q^{(p)}$ given by universal property. In local coordinates $F$ is given by $ X_i\to X_i^p$. Note that it is not a bundle map but it preserves fibers and sends $Q0$ to $Q^{(p)}0$ so it induces a "homomorphism" ( I am not sure in what sense) in etale topology which is purely inseparable. Then Quillen concludes that $(Q0)_{et} \simeq (Q^{(p)}0) _{et} $. Can someone help me understand why these two have same etale homotopy type?
1 Answer
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The claim is that F: Q0 > Q^(p)0 induces, on pullback, an equivalence on etale sites (and hence on etale homotopy types). In fact, more generally this holds for any "universal homeomorphism" between schemes  this is the socalled "topological invariance of the etale site", proved in SGA somewhere.

$\begingroup$ Could you elaborate on how purely inseparability gives equivalence of etale sites? $\endgroup$ Apr 30, 2012 at 7:57

$\begingroup$ Sam: it looks like section 45 of math.columbia.edu/algebraic_geometry/stacksgit/… has it. $\endgroup$ Apr 30, 2012 at 13:27