# The Stacks Project

## Tag 0APF

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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