The Stacks project

72.6 Syntomic topology

In this section we discuss the notion of a syntomic covering of algebraic spaces, and we define the big syntomic site of an algebraic space. Please compare with Topologies, Section 34.6.

Definition 72.6.1. Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. A syntomic covering of $X$ is a family of morphisms $\{ f_ i : X_ i \to X\} _{i \in I}$ of algebraic spaces over $S$ such that each $f_ i$ is syntomic and such that

\[ |X| = \bigcup \nolimits _{i \in I} |f_ i|(|X_ i|), \]

i.e., the morphisms are jointly surjective.

This is exactly the same as Topologies, Definition 34.6.1. In particular, if $X$ and all the $X_ i$ are schemes, then we recover the usual notion of a syntomic covering of schemes.

Lemma 72.6.2. Any smooth covering is a syntomic covering, and a fortiori, any ├ętale or Zariski covering is a syntomic covering.

Proof. This is clear from the definitions and the fact that a smooth morphism is syntomic (Morphisms of Spaces, Lemma 66.37.8), and Lemma 72.5.2. $\square$

Lemma 72.6.3. 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. $\square$

To be continued...


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