56.23 Examples of sheaves

Let $S$ and $\tau$ be as in Section 56.20. We have already seen that any representable presheaf is a sheaf on $(\mathit{Sch}/S)_\tau$ or $S_\tau$, see Lemma 56.15.8 and Remark 56.15.9. Here are some special cases.

Definition 56.23.1. On any of the sites $(\mathit{Sch}/S)_\tau$ or $S_\tau$ of Section 56.20.

1. 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.

2. 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.

3. 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 56.17.4 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 38.4. 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 38.5.6 and the following remark.

Remark 56.23.2. Let $G$ be an abstract group. On any of the sites $(\mathit{Sch}/S)_\tau$ or $S_\tau$ of Section 56.20 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 38.5.6. By Lemma 56.15.8 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 56.23.3. 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 56.23.4. In the terminology introduced above a special case of Theorem 56.22.4 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 $X$.

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