## 59.23 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.

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 $S$.

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