(OK when you're of finite krull dimension)

@Girish: the étale topology for smooth schemes over a Noetherian scheme S need not be hypercomplete, even when S is a spec of a field. Nisnevich is OK.

@Girish: this is almost by definition. Being hypercomplete means that Cech descent implies descent with respect to hypercovers, and hence the "S-local equivalences" (and cofibrations) will be the same in both model structures, so the two model structures will coincide.

Look in the proof in the reference I gave you. The key place that the condition is used is if $f:A \to X$ and $B \to X$, then since $X$ is weakly Hausdorff, the fibered product $A \times_X B \subseteq A \times B$ is a closed subset.

@ToddTrimble: I understand your skepticism, since yes, compactly generated (weak) Hausdorff space is not locally Cartesian closed, lets call this category $CG(W)H$. Fix an object $X.$ I believe the main reason $CG(W)H/X$ is not Cartesian closed is that taking the (fibered) product with a map $Y \to X,$ does not preserve colimits. $CG(W)H/X$ is cocomplete, but the colimits are terrible, since if the naive colimit is not Hausdorff, one has to "Hausdorffify" it. The colimits in $CG/X$ however are much more natural and I believe the issue goes away.

@ToddTrimble: I think it's here: link.springer.com/chapter/10.1007%2FBFb0064694

@user40276: left kan extensions along the Yoneda embedding into presheaves always preserve colimits, as they have a right adjoint. In this case the right adjoint is given by sending a LRS to its functor of points. As such a functor of points is always a sheaf, we even have a adjunction between Zariski sheaves and LRS.

I fail to see how the argument is circular. The Zariski site on geometric schemes being subcanonical just means that every scheme is the colimit of the Cech nerve of any Zariski cover, which follows almost immediately from the fact a topological space is a colimit of the Cech nerve of any open cover. Note that even if the Zariski cover is by affines, the Cech nerve need not consist of affines, so this is a weaker statement.

Hi Henrich. This is just a redrafting of what I explained in my answer. The colimit formula is the same.

Well, since every scheme can be covered by affine schemes, there is an equivalence of categories between sheaves on the Zariski site of (geometric) schemes and sheaves on the Zariski site of affine schemes (by the comparison lemma for sites), and both sites are subcanonical. So hence one has the geometric schemes embed fully faithfully into sheaves on affine schemes. And then one can use the above argument.