Let $C$ be a small site with fibre products. The (injective) Čech model structure on simplicial presheaves $\operatorname{sPre}(C)$ on $C$ presents an $(\infty,1)$-topos and one may ask if this $(\infty,1)$-topos is *hypercomplete* or not.

If I understand correctly, hypercompleteness means in this setting precisely that a Bousfield localization of the Čech model structure on $\operatorname{sPre}(C)$ at hypercovers (which results in Jardine's local (injective) model structure) doesn't change anything. In other words, the Čech model structure is already Jardine's local model structure.

Let $X$ be a scheme and $\mathcal{F}$ a quasi-coherent sheaf on $X$ with values in abelian groups. Sometimes, the Zariski-sheaf cohomology $H^n(X,\mathcal{F})$ of $X$ with values in $\mathcal {F}$ can be calculated by Čech cohomology since a certain spectral sequence degenerates. This is the case for example when $X$ is separated.

I hope that I recall it correctly but there seem to be results (stacks project, TAG 01H0) that this is true for a general $X$, if one considers ''Hypercover-Čech cohomology'', i.e. Čech cohomology with respect to hypercovers instead of just ordinary covers. For example, if $X$ is not separated, the intersection of two open affines does not have to be affine - but it is covered by affines, and so on.

Let $X_{Zar}$ denote the small Zariski-site on a scheme $X$.

What is the relation between the $(\infty,1)$-topos associated to $X_{Zar}$ being hypercomplete and the property that sheaf cohomology can be calculated by ordinary Čech cohomology?

Hypercovers and simplicial presheavesof Dugger et alii? They show in example A.10 that in general there isn't hypercompleteness. But the base site is pathological, so it doesn't relate directly with your question. $\endgroup$