# The Stacks Project

## Tag: 024C

This tag has label descent-lemma-cartesian-equivalent-descent-datum and it points to

The corresponding content:

Lemma 31.36.5. Let $f : X \to S$ be a morphism of schemes. The construction $$\begin{matrix} \text{category of cartesian } \\ \text{schemes over } (X/S)_\bullet \end{matrix} \longrightarrow \begin{matrix} \text{ category of descent data} \\ \text{ relative to } X/S \end{matrix}$$ of Lemma 31.36.4 is an equivalence of categories.

Proof. The functor from left to right is given in Lemma 31.36.4. Hence this is a special case of Lemma 31.36.2. $\square$

\begin{lemma}
\label{lemma-cartesian-equivalent-descent-datum}
Let $f : X \to S$ be a morphism of schemes. The construction
$$\begin{matrix} \text{category of cartesian } \\ \text{schemes over } (X/S)_\bullet \end{matrix} \longrightarrow \begin{matrix} \text{ category of descent data} \\ \text{ relative to } X/S \end{matrix}$$
of Lemma \ref{lemma-cartesian-over}
is an equivalence of categories.
\end{lemma}

\begin{proof}
The functor from left to right is given in
Lemma \ref{lemma-cartesian-over}.
Hence this is a special case of
Lemma \ref{lemma-characterize-cartesian-schemes}.
\end{proof}


To cite this tag (see How to reference tags), use:

\cite[\href{http://stacks.math.columbia.edu/tag/024C}{Tag 024C}]{stacks-project}


In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).