Lemma 69.18.1. Let $S = U \cup W$ be an open covering of a scheme. Then the functor

given by base change is an equivalence where $FP_ T$ is the category of algebraic spaces of finite presentation over the scheme $T$.

Lemma 69.18.1. Let $S = U \cup W$ be an open covering of a scheme. Then the functor

\[ FP_ S \longrightarrow FP_ U \times _{FP_{U \cap W}} FP_ W \]

given by base change is an equivalence where $FP_ T$ is the category of algebraic spaces of finite presentation over the scheme $T$.

**Proof.**
First, since $S = U \cup W$ is a Zariski covering, we see that the category of sheaves on $(\mathit{Sch}/S)_{fppf}$ is equivalent to the category of triples $(\mathcal{F}_ U, \mathcal{F}_ W, \varphi )$ where $\mathcal{F}_ U$ is a sheaf on $(\mathit{Sch}/U)_{fppf}$, $\mathcal{F}_ W$ is a sheaf on $(\mathit{Sch}/W)_{fppf}$, and

\[ \varphi : \mathcal{F}_ U|_{(\mathit{Sch}/U \cap W)_{fppf}} \longrightarrow \mathcal{F}_ W|_{(\mathit{Sch}/U \cap W)_{fppf}} \]

is an isomorphism. See Sites, Lemma 7.26.5 (note that no other gluing data are necessary because $U \times _ S U = U$, $W \times _ S W = W$ and that the cocycle condition is automatic for the same reason). Now, if the sheaf $\mathcal{F}$ on $(\mathit{Sch}/S)_{fppf}$ maps to $(\mathcal{F}_ U, \mathcal{F}_ W, \varphi )$ via this equivalence, then $\mathcal{F}$ is an algebraic space if and only if $\mathcal{F}_ U$ and $\mathcal{F}_ W$ are algebraic spaces. This follows immediately from Algebraic Spaces, Lemma 64.8.5 as $\mathcal{F}_ U \to \mathcal{F}$ and $\mathcal{F}_ W \to \mathcal{F}$ are representable by open immersions and cover $\mathcal{F}$. Finally, in this case the algebraic space $\mathcal{F}$ is of finite presentation over $S$ if and only if $\mathcal{F}_ U$ is of finite presentation over $U$ and $\mathcal{F}_ W$ is of finite presentation over $W$ by Morphisms of Spaces, Lemmas 66.8.8, 66.4.12, and 66.28.4. $\square$

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)