The Stacks project

Maybe it's also worth mentioning that the proof that formally étale is local on the target is easy (so we don't need the highly non-trivial Raynaud-Gruson's results involved in the analogous result for formally smooth). One direction is Lemma 37.8.5. Conversely, suppose $f:X\to S$ is a morphism of schemes for which there is an open cover $S=\bigcup U_i$ such that $f^{-1}U_i\to U_i$ is formally étale for every $i$. Consider a solid diagram as in Definition 37.8.1. Let $V_i'$ be the inverse image of $U_i$ in $T'$ and let $V_i=i^{-1}V_i'$. Then $V_i\to V_i'$ is a first order thickening of schemes. Thus, by Lemma 37.8.2, there is a unique $U_i$-morphism $V_i'\to f^{-1}U_i$ factoring $V_i\to f^{-1}U_i$. It follows that the morphisms $V_i'\to f^{-1}U_i$ glue to an $S$-morphism $T'\to X$ factoring $T\to X$, by the uniqueness of Lemma 37.8.2 ($V_i\cap V_j\to V_i'\cap V_j'$ is also a first order thickening). We get uniqueness of $T'\to X$ from the uniqueness when restricting the diagram to $U_i$.

You may want to quote Lemma 099M for the first assertion of (1).

In the fourth paragraph in the proof, it should be $H_{etale}^i(Z,G)$ instead of $H_{etale}^i(Z,F)$.

Later on, when the vanishing bound on $H^i(X,Q)$ is established, the proof could cite the spectral sequence of hypercohomology.

On Line 26, "pusforward" should be "pushforward".

One could add to this section the result that "being formally étale" is a property that is both local on the source and on the target (i.e., the result analogous to Lemma 37.11.10 by replacing "formally smooth" by "formally étale"). This follows from Lemma 37.11.10 and #11287.

One could add to this section the result that "being formally unramified" is a property that is both local on the source and on the target (i.e., the result analogous to Lemma 37.11.10 by replacing "formally smooth" by "formally unramified"). This follows by combining Lemma 37.6.7 and Morphisms, Lemma 29.32.3.

Shouldn't one invoke Lemma 28.21.2 too? From the assumption that $\mathcal{G}(V)$ is $\mathcal{O}_Y(V)$-projective for every affine $V\subset Y$, Algebra, Lemma 10.94.1 only gives an affine cover $X=\bigcup U_i$ such that $(f^*\mathcal{G})(U_i)$ is $\mathcal{O}_X(U_i)$-projective.

The proof of More on Morphisms, Lemma 37.11.10 cites Definition 20.4.1 for the definition of "pseudo torsor", but the latter only defines torsors, not pseudo torsors.

The target of $\beta$ should be $t$.

nice project, however i take the knowledge.

I think the F_p-dimension of mod p etale cohomology H^q is only not greater than the k-dimension of H^q(X, O_X), they are not always equal as stated in the proof.

Typo in second paragraph of Second proof. "The map $h_n \colon V_n \times \mathrm{Mo}r_{\Delta}([k],[1])$ ..." should be $\mathrm{Mo}r_{\Delta}([n],[1])$.

Typo in second paragraph of Second proof. "The map $h_n \colon V_n \times \Mor_{\Delta}(*[k]*, [1])$ ..." should be $\Mor_{\Delta}([n],[1])$.

The second sentence of Lemma 31.14.5 should be "Let $D$ be an effective Cartier divisor on $S$".

I agree with Torsten, except I would write "largest open subspace" instead of "unique maximal open subspace".

One could make the statement slightly more precise by saying that there exists a unique maximal open subspace $U \subset X$ such that ... Similarly, for the analogous statement for algebraic stacks (Tag 0DZR).

Two typos in the proof:

"The invertible sheaf $\mathcal{L}$ is the restriction of $\mathcal{O}_{\overline{X}'}(n)$ to $X$"

should be replaced by

"The invertible sheaf $\mathcal{L}^n$ is the restriction of $\mathcal{O}_{\overline{X}'}(n)$ to $X'$".

In the first proof, what are $S_f$ and $S_{\mathfrak{p}}$? I thought we had $f \in R$ and $\mathfrak{p} \subseteq R$?

You may want to say what $I$ is in the statement.

In the first line of the description of $H^1_{etale}(X,\mu_n)$, $/\cong$ should not be there and instead $\alpha$ should be an isomorphism.