# The Stacks Project

## Tag: 041B

This tag has label spaces-topologies-lemma-syntomic and it points to

The corresponding content:

Lemma 51.5.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.
1. If $X' \to X$ is an isomorphism then $\{X' \to X\}$ is a syntomic covering of $X$.
2. If $\{X_i \to X\}_{i\in I}$ is a syntomic covering and for each $i$ we have a syntomic covering $\{X_{ij} \to X_i\}_{j\in J_i}$, then $\{X_{ij} \to X\}_{i \in I, j\in J_i}$ is a syntomic covering.
3. If $\{X_i \to X\}_{i\in I}$ is a syntomic covering and $X' \to X$ is a morphism of algebraic spaces then $\{X' \times_X X_i \to X'\}_{i\in I}$ is a syntomic covering.

Proof. Omitted. $\square$

\begin{lemma}
\label{lemma-syntomic}
Let $S$ be a scheme.
Let $X$ be an algebraic space over $S$.
\begin{enumerate}
\item If $X' \to X$ is an isomorphism then $\{X' \to X\}$
is a syntomic covering of $X$.
\item If $\{X_i \to X\}_{i\in I}$ is a syntomic covering and for each
$i$ we have a syntomic covering $\{X_{ij} \to X_i\}_{j\in J_i}$, then
$\{X_{ij} \to X\}_{i \in I, j\in J_i}$ is a syntomic covering.
\item If $\{X_i \to X\}_{i\in I}$ is a syntomic covering
and $X' \to X$ is a morphism of algebraic spaces then
$\{X' \times_X X_i \to X'\}_{i\in I}$ is a syntomic covering.
\end{enumerate}
\end{lemma}

\begin{proof}
Omitted.
\end{proof}


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