## 115.15 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 66.17.

Lemma 115.15.1. 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

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

is injective.

Proof. Immediate consequence of Properties of Spaces, Proposition 66.17.1. $\square$

Lemma 115.15.2. 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 65.12.5. 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 66.17.1. $\square$

Lemma 115.15.3. 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 66.17.1. $\square$

Remark 115.15.4. This remark used to discuss to what extend the original proof of Lemma 115.15.3 (of December 18, 2009) generalizes.

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