Computing the etale cohomology of spheres

Question

$\newcommand\Z{\mathbb{Z}}$ Let $K$ be an algebraically closed field of characteristic $\ne 2$. We have the unit sphere $S^n:~x_0^2 + \ldots + x_n^2 = 1$.

What are the $\Z/2\Z$ cohomology groups of the sphere - $H^i_{et}(S^n,\Z/2\Z)$?

If $K$ is of characteristic 0, then by comparison with the complex numbers we get $$ H^i_{et}(S^n,\Z/2\Z)= \begin{cases} \Z/2\Z, & \text{if}~~ i=0,n \\ 0, & \text{otherwise} \end{cases} $$ and it should be the same result for characteristic $p$.

I am sure that this has been calculated before, but I could not find a reference. So I would like a reference to the calculation of this (or practically equivalent) cohomology groups.

I have 4 different directions to the computation, all of which require some technical details to complete or a smarter decomposition of the space.

1. Comparison for positive characteristic. I understand that there are methods to make a comparison between the etale cohomology over the complex numbers even when the field is of positive characteristic, by viewing the variety as the specialization of a scheme defined over a $p$-adic ring, such that this scheme is nice enough so that there is an equivalence between the cohomology groups over the generic fiber and the cohomology groups of the fiber over the closed point.
2. $S^n$ as a homotopy fiber. The fiber of the simplicial schemes $O_n\backslash\backslash S^n \to BO_{n+1}$ is the simplicial scheme associated with $S^n$. The conditions given by Friedlander for the identification of the cohomology of the sphere with that of the homotopy fiber of the etale homotopy types seems to hold. Jardine calculated the etale cohomology groups of $BO_{n+1}$, and the map above should be the same as the map $BO_n \to BO_{n+1}$. Then a simple spectral sequence argument calculates the right cohomology groups.
3. Using a (hyper)cover. For sphere of odd dimension, which we can write as $z_0 z_1 +\ldots+ z_{n-1} z_n =1$, we can find a cover given by $U_k: z_{2k} \ne 0$. By taking colimit of the etale homotopy type, this cover can be seen to exhibit $S^n$ as the smash product of the $U_k$, each homotopy equivalent to $\mathbb{G}_m$.
4. Gysin Sequence. There are closed embeddings between the spheres $S^{n-1} \to S^n$, which gives rise to a long exact sequence of a pair. We can calculate easily the cohomology with closed support, but I did not find a closed subscheme isomorphic to $S^{n-1}$ such that the complement open subset is easily computable.

Answer 1

This is true over any algebraically closed field $k$ of characteristic different from $2$. More generally, if $\ell$ is invertible in $k$, then $$H^i(X,\mathbb Z_\ell) = \left\{\begin{array}{ll}\mathbb Z_\ell & i = 0, n, \\ 0 & i \neq 0, n. \end{array}\right.$$ See for example [SGA 7$_\text{II}$, Exp. XII, Table 3.7]. The result then follows from the universal coefficient theorem (or you could mimic the proof).

The proof passes through the projective quadric $\bar X$ by removing a hyperplane (isomorphic to the same quadric of dimension $n-1$). The long exact sequence of compactly supported cohomology and induction computes the compactly supported cohomology of $X$. Then Poincaré duality computes the cohomology of $X$.

For the projective quadric, by the Lefschetz hyperplane theorem it suffices to compute the middle cohomology. If you ignore torsion, this is easy: just compute the Euler characteristic (as top Chern class of the tangent bundle). For the statement that $H^n$ is torsion free, you have to a bit more work.

It is a little bit unclear to me what argument SGA uses to prove the torsion-freeness (in the proof of Theorem 3.3, it is not addressed), but it certainly follows from the same result over $\mathbb C$ together with the smooth and proper base change theorems (this is implicitly the strategy of Vérification 3.8).