Sites without a terminal object

Let $\mathcal C$ be a site with a terminal object $X$ . Then the cohomology on the site is defined as the derived functors of the global sections functor $\Gamma(X,-)$ . But what do we do if the site does not have a terminal object?

The solution is to define $H^i(\mathcal C,-)$ as $\Ext{\mathcal O}{i}(\mathcal O,-)$ , where $\mathcal O$ denotes the structure sheaf if $\mathcal C$ is a ringed site. If $\mathcal C$ is not equipped with a ring structure, we take $\mathcal O$ to be the constant sheaf $\underline{\mathbb Z}$ ; this makes $\mathcal C$ into a ringed site.

Lemma. Let $\mathcal C$ be a site with a terminal object $X$ . Then the above definitions agree, i.e.

$H^i(X,-) = \Ext{\mathcal O}{i}(\mathcal O,-).$

Proof. Note that $\Hom_{\mathcal O}(\mathcal O, \mathscr F) = \Gamma(X, \mathscr F)$ , since any map $\mathcal O(X) \to \mathscr F(X)$ can be uniquely extended to a morphism of (pre)sheaves $\mathcal O \to \mathscr F$ , and conversely every such morphism is determined by its map on global sections. The result now follows since $\Ext{\mathcal O}{i}(\mathcal O, -)$ and $H^i(X,-)$ are defined as the derived functors of $\Hom_{\mathcal O}(\mathcal O,-)$ and $\Gamma(X,-)$ respectively. $\qedsymbol$

Remark. From this perspective, it seems quite magical that for a sheaf $\mathscr F$ of $\mathcal O_X$ -modules on a ringed space $(X,\mathcal O_X)$ , the cohomology groups $\Ext{\mathcal O_X}{i}(\mathcal O_X,\mathscr F)$ and $\Ext{\underline{\Z}}{i}(\underline{\Z},\mathscr F)$ 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 $G$ be a group scheme over $k$ . Then we have a stack $BG$ of $G$ -torsors. The objects of $BG$ are pairs $(U,P)$ , where $U$ is a $k$ -scheme and $P$ is a $G$ -torsor over $U$ . Morphisms $(U,P) \to (U',P')$ are pairs $(f,g) \colon (U,P) \to (U',P')$ making the diagram

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commutative. This forces the diagram to be a pullback, since all maps between $G$ -torsors are isomorphisms.

The (large) Zariski site on $BG$ is defined by declaring coverings $\{(U_i, P_i) \to (U,P)\}$ to be families such that $\{U_i \to U\}$ is a Zariski covering (and similarly for the étale and fppf sites).

Now does the category $BG$ have a terminal object? This would be a $G$ -torsor $P_0 \to U_0$ such that every other $G$ -torsor $P \to U$ admits a unique map to it, realising $P$ as the pullback of $P_0$ along $U \to U_0$ . But this object would exactly be the classifying stack $U_0 = BG$ , 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 $X/k$ be a variety in characteristic $p > 0$ ; for simplicity, let’s say $k = \mathbb F_p$ . Then consider the crystalline site of $X/\Spec(\Z/p^n\Z)$ . Roughly speaking, its objects are triples $(U,T,\delta)$ , where $U \to X$ is an open immersion, $U \to T$ is a thickening with a map to $\Spec{\F_p} \to \Spec{\Z/p^n\Z}$ , and $\delta$ is a divided power structure on the ideal sheaf $\mathcal I_U \subseteq \mathcal O_T$ (with a compatibility condition w.r.t. $\Spec{\F_p} \to \Spec{\Z/p^n\Z}$ ). There is a suitable notion of morphisms.

This site does not have a terminal object, basically because there are many thickenings on $U = X$ 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.)

Lovely little lemmas

Or maybe ‘amusing observations’

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