etale homotopy and Adams conjecture

Question

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 $Q-0$ 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 $(Q-0)_{et} \simeq (Q^{(p)}-0) _{et} $. Can someone help me understand why these two have same etale homotopy type?

Answer 1

The claim is that F: Q-0 --> 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 so-called "topological invariance of the etale site", proved in SGA somewhere.