Proposition 98.26.1. Let S be a locally Noetherian scheme. Let F : (\textit{Noetherian}/S)_{\acute{e}tale}^{opp} \to \textit{Sets} be a functor. Assume that
\Delta : F \to F \times F is representable (as a transformation of functors, see Categories, Definition 4.6.4),
F satisfies axioms [-1], [0], [1], [2], [3], [4], [5] (see above), and
\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 98.25. 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 étale topology) and which is limit preserving, see Lemma 98.25.2. Then F' satisfies axioms [0] and [1] as given in Section 98.15. By Lemma 98.25.4 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 98.15 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 98.16.1.
\square
Comments (0)