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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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