The Stacks project

Definition 96.24.2. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids.

  1. The associated affine Zariski site $\mathcal{X}_{affine, Zar}$ is the structure of site on $\mathcal{X}_{affine}$ inherited from $(\textit{Aff}/S)_{Zar}$.

  2. The associated affine étale site $\mathcal{X}_{affine, {\acute{e}tale}}$ is the structure of site on $\mathcal{X}_{affine}$ inherited from $(\textit{Aff}/S)_{\acute{e}tale}$.

  3. The associated affine smooth site $\mathcal{X}_{affine, smooth}$ is the structure of site on $\mathcal{X}_{affine}$ inherited from $(\textit{Aff}/S)_{smooth}$.

  4. The associated affine syntomic site $\mathcal{X}_{affine, syntomic}$ is the structure of site on $\mathcal{X}_{affine}$ inherited from $(\textit{Aff}/S)_{syntomic}$.

  5. The associated affine fppf site $\mathcal{X}_{affine, fppf}$ is the structure of site on $\mathcal{X}_{affine}$ inherited from $(\textit{Aff}/S)_{fppf}$.

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