## 34.11 Change of topologies

Let $f : X \to Y$ be a morphism of schemes over a base scheme $S$. In this case we have the following morphisms of sites1 (with suitable choices of sites as in Remark 34.11.1 below):

1. $(\mathit{Sch}/X)_{fppf} \longrightarrow (\mathit{Sch}/Y)_{fppf}$,

2. $(\mathit{Sch}/X)_{fppf} \longrightarrow (\mathit{Sch}/Y)_{syntomic}$,

3. $(\mathit{Sch}/X)_{fppf} \longrightarrow (\mathit{Sch}/Y)_{smooth}$,

4. $(\mathit{Sch}/X)_{fppf} \longrightarrow (\mathit{Sch}/Y)_{\acute{e}tale}$,

5. $(\mathit{Sch}/X)_{fppf} \longrightarrow (\mathit{Sch}/Y)_{Zar}$,

6. $(\mathit{Sch}/X)_{syntomic} \longrightarrow (\mathit{Sch}/Y)_{syntomic}$,

7. $(\mathit{Sch}/X)_{syntomic} \longrightarrow (\mathit{Sch}/Y)_{smooth}$,

8. $(\mathit{Sch}/X)_{syntomic} \longrightarrow (\mathit{Sch}/Y)_{\acute{e}tale}$,

9. $(\mathit{Sch}/X)_{syntomic} \longrightarrow (\mathit{Sch}/Y)_{Zar}$,

10. $(\mathit{Sch}/X)_{smooth} \longrightarrow (\mathit{Sch}/Y)_{smooth}$,

11. $(\mathit{Sch}/X)_{smooth} \longrightarrow (\mathit{Sch}/Y)_{\acute{e}tale}$,

12. $(\mathit{Sch}/X)_{smooth} \longrightarrow (\mathit{Sch}/Y)_{Zar}$,

13. $(\mathit{Sch}/X)_{\acute{e}tale}\longrightarrow (\mathit{Sch}/Y)_{\acute{e}tale}$,

14. $(\mathit{Sch}/X)_{\acute{e}tale}\longrightarrow (\mathit{Sch}/Y)_{Zar}$,

15. $(\mathit{Sch}/X)_{Zar} \longrightarrow (\mathit{Sch}/Y)_{Zar}$,

16. $(\mathit{Sch}/X)_{fppf} \longrightarrow Y_{\acute{e}tale}$,

17. $(\mathit{Sch}/X)_{syntomic} \longrightarrow Y_{\acute{e}tale}$,

18. $(\mathit{Sch}/X)_{smooth} \longrightarrow Y_{\acute{e}tale}$,

19. $(\mathit{Sch}/X)_{\acute{e}tale}\longrightarrow Y_{\acute{e}tale}$,

20. $(\mathit{Sch}/X)_{fppf} \longrightarrow Y_{Zar}$,

21. $(\mathit{Sch}/X)_{syntomic} \longrightarrow Y_{Zar}$,

22. $(\mathit{Sch}/X)_{smooth} \longrightarrow Y_{Zar}$,

23. $(\mathit{Sch}/X)_{\acute{e}tale}\longrightarrow Y_{Zar}$,

24. $(\mathit{Sch}/X)_{Zar} \longrightarrow Y_{Zar}$,

25. $X_{\acute{e}tale}\longrightarrow Y_{\acute{e}tale}$,

26. $X_{\acute{e}tale}\longrightarrow Y_{Zar}$,

27. $X_{Zar} \longrightarrow Y_{Zar}$,

In each case the underlying continuous functor $\mathit{Sch}/Y \to \mathit{Sch}/X$, or $Y_\tau \to \mathit{Sch}/X$ is the functor $Y'/Y \mapsto X \times _ Y Y'/X$. Namely, in the sections above we have seen the morphisms $f_{big} : (\mathit{Sch}/X)_\tau \to (\mathit{Sch}/Y)_\tau$ and $f_{small} : X_\tau \to Y_\tau$ for $\tau$ as above. We also have seen the morphisms of sites $\pi _ Y : (\mathit{Sch}/Y)_\tau \to Y_\tau$ for $\tau \in \{ {\acute{e}tale}, Zariski\}$. On the other hand, it is clear that the identity functor $(\mathit{Sch}/X)_\tau \to (\mathit{Sch}/X)_{\tau '}$ defines a morphism of sites when $\tau$ is a stronger topology than $\tau '$. Hence composing these gives the list of possible morphisms above.

Because of the simple description of the underlying functor it is clear that given morphisms of schemes $X \to Y \to Z$ the composition of two of the morphisms of sites above, e.g.,

$(\mathit{Sch}/X)_{\tau _0} \longrightarrow (\mathit{Sch}/Y)_{\tau _1} \longrightarrow (\mathit{Sch}/Z)_{\tau _2}$

is the corresponding morphism of sites associated to the morphism of schemes $X \to Z$.

Remark 34.11.1. Take any category $\mathit{Sch}_\alpha$ constructed as in Sets, Lemma 3.9.2 starting with the set of schemes $\{ X, Y, S\}$. Choose any set of coverings $\text{Cov}_{fppf}$ on $\mathit{Sch}_\alpha$ as in Sets, Lemma 3.11.1 starting with the category $\mathit{Sch}_\alpha$ and the class of fppf coverings. Let $\mathit{Sch}_{fppf}$ denote the big fppf site so obtained. Next, for $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic\}$ let $\mathit{Sch}_\tau$ have the same underlying category as $\mathit{Sch}_{fppf}$ with coverings $\text{Cov}_\tau \subset \text{Cov}_{fppf}$ simply the subset of $\tau$-coverings. It is straightforward to check that this gives rise to a big site $\mathit{Sch}_\tau$.

[1] We have not included the comparison between the ph topology and the others; for this see More on Morphisms, Remark 37.48.8.

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