Section 59.23 (03YZ): Examples of sheaves

Let

$S$ and

$\tau$ be as in Section 59.20. We have already seen that any representable presheaf is a sheaf on

$(\mathit{Sch}/S)_\tau$ or

$S_\tau$ , see Lemma 59.15.8 and Remark 59.15.9. Here are some special cases.

Definition 59.23.1. On any of the sites $(\mathit{Sch}/S)_\tau$ or $S_\tau$ of Section 59.20.

The sheaf $T \mapsto \Gamma (T, \mathcal{O}_ T)$ is denoted $\mathcal{O}_ S$ , or $\mathbf{G}_ a$ , or $\mathbf{G}_{a, S}$ if we want to indicate the base scheme.
Similarly, the sheaf $T \mapsto \Gamma (T, \mathcal{O}^*_ T)$ is denoted $\mathcal{O}_ S^*$ , or $\mathbf{G}_ m$ , or $\mathbf{G}_{m, S}$ if we want to indicate the base scheme.
The constant sheaf $\underline{\mathbf{Z}/n\mathbf{Z}}$ on any site is the sheafification of the constant presheaf $U \mapsto \mathbf{Z}/n\mathbf{Z}$ .

The first is a sheaf by Theorem 59.17.4 for example. The second is a sub presheaf of the first, which is easily seen to be a sheaf itself. The third is a sheaf by definition. Note that each of these sheaves is representable. The first and second by the schemes

$\mathbf{G}_{a, S}$ and

$\mathbf{G}_{m, S}$ , see Groupoids, Section 39.4. The third by the finite étale group scheme

$\mathbf{Z}/n\mathbf{Z}_ S$ sometimes denoted

$(\mathbf{Z}/n\mathbf{Z})_ S$ which is just

$n$ copies of

$S$ endowed with the obvious group scheme structure over

$S$ , see Groupoids, Example 39.5.6 and the following remark.

Remark 59.23.2. Let $G$ be an abstract group. On any of the sites $(\mathit{Sch}/S)_\tau$ or $S_\tau$ of Section 59.20 the sheafification $\underline{G}$ of the constant presheaf associated to $G$ in the Zariski topology of the site already gives

$\Gamma (U, \underline{G}) = \{ \text{Zariski locally constant maps }U \to G\}$

This Zariski sheaf is representable by the group scheme $G_ S$ according to Groupoids, Example 39.5.6. By Lemma 59.15.8 any representable presheaf satisfies the sheaf condition for the $\tau$ -topology as well, and hence we conclude that the Zariski sheafification $\underline{G}$ above is also the $\tau$ -sheafification.

Definition 59.23.3. Let $S$ be a scheme. The structure sheaf of $S$ is the sheaf of rings $\mathcal{O}_ S$ on any of the sites $S_{Zar}$ , $S_{\acute{e}tale}$ , or $(\mathit{Sch}/S)_\tau$ discussed above.

If there is some possible confusion as to which site we are working on then we will indicate this by using indices. For example we may use

$\mathcal{O}_{S_{\acute{e}tale}}$ to stress the fact that we are working on the small étale site of

$S$ .

Remark 59.23.4. In the terminology introduced above a special case of Theorem 59.22.4 is

$H_{fppf}^ p(X, \mathbf{G}_ a) = H_{\acute{e}tale}^ p(X, \mathbf{G}_ a) = H_{Zar}^ p(X, \mathbf{G}_ a) = H^ p(X, \mathcal{O}_ X)$

for all $p \geq 0$ . Moreover, we could use the notation $H^ p_{fppf}(X, \mathcal{O}_ X)$ to indicate the cohomology of the structure sheaf on the big fppf site of $X$ .

The Stacks project

59.23 Examples of sheaves

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