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98.26 Algebraic spaces in the Noetherian setting

Let $S$ be a locally Noetherian scheme. Let $(\textit{Noetherian}/S)_{\acute{e}tale}\subset (\mathit{Sch}/S)_{\acute{e}tale}$ denote the site studied in Section 98.25. Let $F : (\textit{Noetherian}/S)_{\acute{e}tale}^{opp} \to \textit{Sets}$ be a functor, i.e., $F$ is a presheaf on $(\textit{Noetherian}/S)_{\acute{e}tale}$. In this setting all the axioms [-1], [0], [1], [2], [3], [4], [5] of Section 98.15 make sense. We will review them one by one and make sure the reader knows exactly what we mean.

Axiom [-1]. This is a set theoretic condition to be ignored by readers who are not interested in set theoretic questions. It makes sense for $F$ since it concerns the evaluation of $F$ on spectra of fields of finite type over $S$ which are objects of $(\textit{Noetherian}/S)_{\acute{e}tale}$.

Axiom [0]. This is the axiom that $F$ is a sheaf on $(\textit{Noetherian}/S)_{\acute{e}tale}^{opp}$, i.e., satisfies the sheaf condition for ├ętale coverings.

Axiom [1]. This is the axiom that $F$ is limit preserving as defined in Section 98.25: for any directed limit of affine schemes $X = \mathop{\mathrm{lim}}\nolimits X_ i$ of $(\textit{Noetherian}/S)_{\acute{e}tale}$ we have $F(X) = \mathop{\mathrm{colim}}\nolimits F(X_ i)$.

Axiom [2]. This is the axiom that $F$ satisfies the Rim-Schlessinger condition (RS). Looking at the definition of condition (RS) in Definition 98.5.1 and the discussion in Section 98.15 we see that this means: given any pushout $Y' = Y \amalg _ X X'$ of schemes of finite type over $S$ where $Y, X, X'$ are spectra of Artinian local rings, then

\[ F(Y \amalg _ X X') \to F(Y) \times _{F(X)} F(X') \]

is a bijection. This condition makes sense as the schemes $X$, $X'$, $Y$, and $Y'$ are in $(\text{Noetherian}/S)_{\acute{e}tale}$ since they are of finite type over $S$.

Axiom [3]. This is the axiom that every tangent space $TF_{k, x_0}$ is finite dimensional. This makes sense as the tangent spaces $TF_{k, x_0}$ are constructed from evaluations of $F$ at $\mathop{\mathrm{Spec}}(k)$ and $\mathop{\mathrm{Spec}}(k[\epsilon ])$ with $k$ a field of finite type over $S$ and hence are obtained by evaluating at objects of the category $(\textit{Noetherian}/S)_{\acute{e}tale}$.

Axiom [4]. This is axiom that the every formal object is effective. Looking at the discussion in Sections 98.9 and 98.15 we see that this involves evaluating our functor at Noetherian schemes only and hence this condition makes sense for $F$.

Axiom [5]. This is the axiom stating that $F$ satisfies openness of versality. Recall that this means the following: Given a scheme $U$ locally of finite type over $S$, given $x \in F(U)$, and given a finite type point $u_0 \in U$ such that $x$ is versal at $u_0$, then there exists an open neighbourhood $u_0 \in U' \subset U$ such that $x$ is versal at every finite type point of $U'$. As before, verifying this only involves evaluating our functor at Noetherian schemes.

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

  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 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 ├ętale 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. $\square$

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