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.)