I would like to study the ordered Cech cohomology with respect to a Zariski covering of a variety. I can pass to the limit with respect to refinements; the components of the 'limit covering' will be local schemes. Unfortunately, it seems that the intersection of two components of this type is not local. Is there a way to fix that? I would be satisfied with replacing the Cech cohomology by the cohomology with respect to 'something like a Zariski hypercovering'; here I need this to be similar to the ordered Cech cohomology.
Does the situation become nicer if we consider ordered Cech cohomology with respect to certain Nisnevich covers (there are cases when ordered Cech cohomology is reasonable; at least this is so for the so-called elementary distinguished squares)?