## Tag `0APF`

Chapter 38: Groupoid Schemes > Section 38.21: Descent in terms of groupoids

Lemma 38.21.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of schemes over $S$. The construction of Lemma 38.21.2 determines an equivalence $$ \begin{matrix} \text{category of groupoid schemes} \\ \text{cartesian over } (X, X \times_Y X, \ldots) \end{matrix} \longrightarrow \begin{matrix} \text{ category of descent data} \\ \text{ relative to } X/Y \end{matrix} $$

Proof.This is clear from Lemma 38.21.2 and the definition of descent data on schemes in Descent, Definition 34.31.1. $\square$

The code snippet corresponding to this tag is a part of the file `groupoids.tex` and is located in lines 3803–3819 (see updates for more information).

```
\begin{lemma}
\label{lemma-cartesian-equivalent-descent-datum}
Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of schemes over $S$.
The construction of Lemma \ref{lemma-characterize-cartesian-schemes}
determines an equivalence
$$
\begin{matrix}
\text{category of groupoid schemes} \\
\text{cartesian over } (X, X \times_Y X, \ldots)
\end{matrix}
\longrightarrow
\begin{matrix}
\text{ category of descent data} \\
\text{ relative to } X/Y
\end{matrix}
$$
\end{lemma}
\begin{proof}
This is clear from
Lemma \ref{lemma-characterize-cartesian-schemes}
and the definition of descent data on schemes in
Descent, Definition \ref{descent-definition-descent-datum}.
\end{proof}
```

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