TITLE: equivalence of Grothendieck-style versus Cech-style sheaf cohomology
QUESTION [95 upvotes]: Given a topological space $X$, we can define the sheaf cohomology of $X$ in 
I. the Grothendieck style (as the right derived functor of the global sections functor $\Gamma(X,-)$)
or
II. the Čech style (first by defining the Čech cohomology groups subordinate to an open cover, and then taking the direct limit of these groups over all covers).
When exactly are these two definitions equivalent? I'm unhappy with the explanation given by Hartshorne. Are they the same for any paracompact Hausdorff space? Or a locally contractible space?
And what is the relationship between these two sheaf cohomologies and singular cohomology?
Any elaboration on this circle of ideas related to the relationship between all the different cohomology theories would be appreciated.

REPLY [4 votes]: 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.