## 114.14 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 65.17.

Lemma 114.14.1. 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 65.17.1. $\square$

Lemma 114.14.2. 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 64.12.5. 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 65.17.1. $\square$

Lemma 114.14.3. 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 65.17.1. $\square$

Remark 114.14.4. This remark used to discuss to what extend the original proof of Lemma 114.14.3 (of December 18, 2009) generalizes.

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 toolbar).