## Tag `07IK`

Chapter 51: Crystalline Cohomology > Section 51.9: The crystalline site

Remark 51.9.3 (Functoriality). Let $p$ be a prime number. Let $(S, \mathcal{I}, \gamma) \to (S', \mathcal{I}', \gamma')$ be a morphism of divided power schemes over $\mathbf{Z}_{(p)}$. Let $$ \xymatrix{ X \ar[r]_f \ar[d] & Y \ar[d] \\ S_0 \ar[r] & S'_0 } $$ be a commutative diagram of morphisms of schemes and assume $p$ is locally nilpotent on $X$ and $Y$. By analogy with Topologies, Lemma 33.3.16 we define $$ f_{\text{cris}} : (X/S)_{\text{cris}} \longrightarrow (Y/S')_{\text{cris}} $$ by the formula $f_{\text{cris}} = \pi_Y \circ f_{\text{CRIS}} \circ i_X$ where $i_X$ and $\pi_Y$ are as in Lemma 51.9.2 for $X$ and $Y$ and where $f_{\text{CRIS}}$ is as in Remark 51.8.5.

The code snippet corresponding to this tag is a part of the file `crystalline.tex` and is located in lines 1910–1932 (see updates for more information).

```
\begin{remark}[Functoriality]
\label{remark-functoriality-cris}
Let $p$ be a prime number.
Let $(S, \mathcal{I}, \gamma) \to (S', \mathcal{I}', \gamma')$
be a morphism of divided power schemes over $\mathbf{Z}_{(p)}$.
Let
$$
\xymatrix{
X \ar[r]_f \ar[d] & Y \ar[d] \\
S_0 \ar[r] & S'_0
}
$$
be a commutative diagram of morphisms of schemes and assume $p$ is
locally nilpotent on $X$ and $Y$. By analogy with
Topologies, Lemma \ref{topologies-lemma-morphism-big-small} we define
$$
f_{\text{cris}} : (X/S)_{\text{cris}} \longrightarrow (Y/S')_{\text{cris}}
$$
by the formula $f_{\text{cris}} = \pi_Y \circ f_{\text{CRIS}} \circ i_X$
where $i_X$ and $\pi_Y$ are as in Lemma \ref{lemma-compare-big-small}
for $X$ and $Y$ and where $f_{\text{CRIS}}$ is as in
Remark \ref{remark-functoriality-big-cris}.
\end{remark}
```

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