Here are some positive results and counterexamples for etale cohomology.

**Definition**: Let $X$ be a scheme. Say that $X$ has property "$AF_{n}$" if for every collection $x_{1},\dotsc,x_{n} \in X$ of $n$ points of $X$ there exists an affine open subscheme $U \subseteq X$ containing all the $x_{i}$. Let $$ a(X) $$ denote the supremum of the positive integers $n$ for which $X$ has $AF_{n}$. We say that $X$ is an "AF-scheme" (or "FA-scheme") if $a(X) = \infty$.

See Gross [4], Section 2.

The relevant theorem is:

**Theorem** (Artin 1971 [1]): Let $X$ be a quasi-compact AF-scheme. Then for any abelian sheaf $\mathscr{F}$ on the etale site of $X$, the map $$ \check{\mathrm{H}}^{i}(X,\mathscr{F}) \to \mathrm{H}_{\mathrm{et}}^{i}(X,\mathscr{F}) $$ is an isomorphism for all $i$.

Milne [6, III, Theorem 2.17] also discusses Artin's proof. This is more a claim about the etale topology on $X$, namely that if $U \to X$ is an etale cover, then any etale cover $V \to U \times_{X} \dotsb \times_{X} U$ may be refined by an etale cover of the form $U' \times_{X} \dotsb \times_{X} U'$ for $U' \to X$ which refines $U \to X$.

Later Schröer refined Artin's result as follows:

**Theorem** (Schröer 2003 [7]): Let $X$ be a Noetherian scheme and let $n$ be an integer such that $n \le a(X)$. Then for any abelian sheaf $\mathscr{F}$ on the etale site of $X$, the map $$ \check{\mathrm{H}}^{i}(X,\mathscr{F}) \to \mathrm{H}_{\mathrm{et}}^{i}(X,\mathscr{F}) $$ is an isomorphism for $i \le n$ and an injection for $i = n+1$.

**Example**: For an example of a scheme $X$ and an abelian sheaf $\mathscr{F}$ on the etale site of $X$ for which the map $$ \check{\mathrm{H}}^{2}(X,\mathscr{F}) \to \mathrm{H}_{\mathrm{et}}^{2}(X,\mathscr{F}) $$ is not an isomorphism, see the answers to this question. The current answers discuss the cases (1) $\mathscr{F}$ is a constant sheaf and (2) $\mathscr{F} = \mathcal{O}_{X}$.

(Gabber) For an example of a scheme $X$ for which the map $$ \check{\mathrm{H}}^{2}(X,\mathbb{G}_{m}) \to \mathrm{H}_{\mathrm{et}}^{2}(X,\mathbb{G}_{m}) $$ is not an isomorphism, let $R$ be a normal noetherian strictly henselian local ring of dimension $\ge 2$ whose punctured spectrum $U$ has nonzero Picard group (see e.g. this and this for examples of such $R$), and let $X$ be the gluing of two copies of $\operatorname{Spec} R$ along $U$. This is a local version of the counterexample to $\operatorname{Br} = \operatorname{Br}'$ due to Edidin, Hassett, Kresch, Vistoli [9].

Here are some conditions relevant to the AF-property:

**Lemma** (e.g. [8, 01ZY]): Let $X$ be a quasi-compact scheme admitting an ample line bundle. Then $X$ is an AF-scheme.

This is essentially the graded prime avoidance lemma.

The Chevalley conjecture, proved by Kleiman, states that for smooth proper varieties the converse is true:

**Theorem** (Kleiman 1966 [5]): Let $k$ be an algebraically closed field and let $X$ be a smooth proper $k$-scheme. If $X$ is an AF-scheme, then $X$ is projective over $k$.

Benoist proved the following generalization of Kleiman's result:

**Theorem** (Benoist 2013 [2]): Let $k$ be an algebraically closed field and let $X$ be a normal, finite type $k$-scheme. If $X$ is an AF-scheme, then $X$ is quasi-projective over $k$.

**Theorem** (Farnik 2013 [3, Theorem 2.2]) For every integer $n \ge 2$ there is a smooth proper variety $X$ with $a(X) = n$.

*References*:

[1] M. Artin, "On the joins of Hensel rings", Advances in Mathematics **7** (1971) pp 282–296, doi:10.1016/S0001-8708(71)80007-5, core.ac.uk.

[2] O. Benoist, "Quasi-projectivity of normal varieties", International Mathematics Research Notices, vol 2013, no 17 (2012) pp 3878–3885, doi:10.1093/imrn/rns163, arXiv:1112.0975.

[3] M. Farnik, "On strengthening of the Kleiman-Chevalley criterion", Proceedings of the AMS **141** no 11 (2013) pp 4005-4013, doi:10.1090/S0002-9939-2013-11695-3.

[4] P. Gross, "Tensor generators on schemes and stacks", Algebraic Geometry **4** (4) (2017) pp 501–522, doi:10.14231/ag-2017-026, arXiv:1306.5418.

[5] S. Kleiman, "Toward a numerical theory of ampleness", Annals of Mathematics **84** No. 3 (1966) pp 293–344, doi:10.2307/1970447.

[6] J.S. Milne, Etale Cohomology, Princeton University Press (1980) JSTOR (subscription needed).

[7] S. Schröer, "The bigger Brauer group is really big", Journal of Algebra **262** (2003) pp 210–225, doi:10.1016/S0021-8693(03)00026-7, arXiv:math/0108135.

[8] Stacks Project, https://stacks.math.columbia.edu/.

[9] Edidin, Hassett, Kresch, Vistoli, "Brauer groups and quotient stacks", American Journal of Mathematics **123** No. 4 (2001), doi:10.1353/ajm.2001.0024, JSTOR, arXiv:math/9905049.