The Stacks project

Proposition 97.26.1. Let $S$ be a locally Noetherian scheme. Let $F : (\textit{Noetherian}/S)_{\acute{e}tale}^{opp} \to \textit{Sets}$ be a functor. Assume that

  1. $\Delta : F \to F \times F$ is representable (as a transformation of functors, see Categories, Definition 4.6.4),

  2. $F$ satisfies axioms [-1], [0], [1], [2], [3], [4], [5] (see above), and

  3. $\mathcal{O}_{S, s}$ is a G-ring for all finite type points $s$ of $S$.

Then there exists a unique algebraic space $F' : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ whose restriction to $(\textit{Noetherian}/S)_{\acute{e}tale}$ is $F$ (see proof for elucidation).

Proof. Recall that the sites $(\mathit{Sch}/S)_{fppf}$ and $(\mathit{Sch}/S)_{\acute{e}tale}$ have the same underlying category, see discussion in Section 97.25. Similarly the sites $(\textit{Noetherian}/S)_{\acute{e}tale}$ and $(\textit{Noetherian}/S)_{fppf}$ have the same underlying categories. By axioms [0] and [1] the functor $F$ is a sheaf and limit preserving. Let $F' : (\mathit{Sch}/S)_{\acute{e}tale}^{opp} \to \textit{Sets}$ be the unique extension of $F$ which is a sheaf (for the ├ętale topology) and which is limit preserving, see Lemma 97.25.2. Then $F'$ satisfies axioms [0] and [1] as given in Section 97.15. By Lemma 97.25.4 we see that $\Delta ' : F' \to F' \times F'$ is representable (by schemes). On the other hand, it is immediately clear that $F'$ satisfies axioms [-1], [2], [3], [4], [5] of Section 97.15 as each of these involves only evaluating $F'$ at objects of $(\textit{Noetherian}/S)_{\acute{e}tale}$ and we've assumed the corresponding conditions for $F$. Whence $F'$ is an algebraic space by Proposition 97.16.1. $\square$

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