Let be a site with a terminal object . Then the cohomology on the site is defined as the derived functors of the global sections functor . But what do we do if the site does not have a terminal object?
The solution is to define as , where denotes the structure sheaf if is a ringed site. If is not equipped with a ring structure, we take to be the constant sheaf ; this makes into a ringed site.
Lemma. Let be a site with a terminal object . Then the above definitions agree, i.e.
Proof. Note that , since any map can be uniquely extended to a morphism of (pre)sheaves , and conversely every such morphism is determined by its map on global sections. The result now follows since and are defined as the derived functors of and respectively.
Remark. From this perspective, it seems quite magical that for a sheaf of -modules on a ringed space , the cohomology groups and agree. It turns out that this is true in the setting of ringed sites as well; see Tag 03FD.
So why is this useful? Let’s give some examples of sites that do not have a terminal object.
Example. Let be a group scheme over . Then we have a stack of -torsors. The objects of are pairs , where is a -scheme and is a -torsor over . Morphisms are pairs making the diagram
commutative. This forces the diagram to be a pullback, since all maps between -torsors are isomorphisms.
The (large) Zariski site on is defined by declaring coverings to be families such that is a Zariski covering (and similarly for the étale and fppf sites).
Now does the category have a terminal object? This would be a -torsor such that every other -torsor admits a unique map to it, realising as the pullback of along . But this object would exactly be the classifying stack , which does not exist as a scheme (or algebraic space). The fact that a terminal object does not exist is the whole reason we need to define it as a stack in the first place!
Example. Let be a variety in characteristic ; for simplicity, let’s say . Then consider the crystalline site of . Roughly speaking, its objects are triples , where is an open immersion, is a thickening with a map to , and is a divided power structure on the ideal sheaf (with a compatibility condition w.r.t. ). There is a suitable notion of morphisms.
This site does not have a terminal object, basically because there are many thickenings on with the respective compatibilities. (I am admittedly no expert, and it could very well be true that this is not 100% correct. However, I am certain that the crystalline site in general does not have a terminal object.)