The Stacks project

113.13 Spaces and fpqc coverings

The material here was made obsolete by Gabber's argument showing that algebraic spaces satisfy the sheaf condition with respect to fpqc coverings. Please visit Properties of Spaces, Section 64.17.

Lemma 113.13.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\{ f_ i : T_ i \to T\} _{i \in I}$ be a fpqc covering of schemes over $S$. Then the map

\[ \mathop{Mor}\nolimits _ S(T, X) \longrightarrow \prod \nolimits _{i \in I} \mathop{Mor}\nolimits _ S(T_ i, X) \]

is injective.

Proof. Immediate consequence of Properties of Spaces, Proposition 64.17.1. $\square$

Lemma 113.13.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $X = \bigcup _{j \in J} X_ j$ be a Zariski covering, see Spaces, Definition 63.12.5. If each $X_ j$ satisfies the sheaf property for the fpqc topology then $X$ satisfies the sheaf property for the fpqc topology.

Proof. This is true because all algebraic spaces satisfy the sheaf property for the fpqc topology, see Properties of Spaces, Proposition 64.17.1. $\square$

Lemma 113.13.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $X$ is Zariski locally quasi-separated over $S$, then $X$ satisfies the sheaf condition for the fpqc topology.

Proof. Immediate consequence of the general Properties of Spaces, Proposition 64.17.1. $\square$

Remark 113.13.4. This remark used to discuss to what extend the original proof of Lemma 113.13.3 (of December 18, 2009) generalizes.


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