70.19 Application to modifications
Using limits we can describe the category of modifications of a decent algebraic space over a closed point in terms of the henselian local ring.
Lemma 70.19.1. Let $S$ be a scheme. Consider a separated étale morphism $f : V \to W$ of algebraic spaces over $S$. Assume there exists a closed subspace $T \subset W$ such that $f^{-1}T \to T$ is an isomorphism. Then, with $W^0 = W \setminus T$ and $V^0 = f^{-1}W^0$ the base change functor
\[ \left\{ \begin{matrix} g : X \to W\text{ morphism of algebraic spaces}
\\ g^{-1}(W^0) \to W^0\text{ is an isomorphism}
\end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} h : Y \to V\text{ morphism of algebraic spaces}
\\ h^{-1}(V^0) \to V^0\text{ is an isomorphism}
\end{matrix} \right\} \]
is an equivalence of categories.
Proof.
Since $V \to W$ is separated we see that $V \times _ W V = \Delta (V) \amalg U$ for some open and closed subspace $U$ of $V \times _ W V$. By the assumption that $f^{-1}T \to T$ is an isomorphism we see that $U \times _ W T = \emptyset $, i.e., the two projections $U \to V$ maps into $V^0$.
Given $h : Y \to V$ in the right hand category, consider the contravariant functor $X$ on $(\mathit{Sch}/S)_{fppf}$ defined by the rule
\[ X(T) = \{ (w, y) \mid w : T \to W,\ y : T \times _{w, W} V \to Y\text{ morphism over }V\} \]
Denote $g : X \to W$ the map sending $(w, y) \in X(T)$ to $w \in W(T)$. Since $h^{-1}V^0 \to V^0$ is an isomorphism, we see that if $w : T \to W$ maps into $W^0$, then there is a unique choice for $h$. In other words $X \times _{g, W} W^0 = W^0$. On the other hand, consider a $T$-valued point $(w, y, v)$ of $X \times _{g, W, f} V$. Then $w = f \circ v$ and
\[ y : T \times _{f \circ v, W} V \longrightarrow V \]
is a morphism over $V$. Consider the morphism
\[ T \times _{f \circ v, W} V \xrightarrow {(v, \text{id}_ V)} V \times _ W V = V \amalg U \]
The inverse image of $V$ is $T$ embedded via $(\text{id}_ T, v) : T \to T \times _{f \circ v, W} V$. The composition $y' = y \circ (\text{id}_ T, v) : T \to Y$ is a morphism with $v = h \circ y'$ which determines $y$ because the restriction of $y$ to the other part is uniquely determined as $U$ maps into $V^0$ by the second projection. It follows that $X \times _{g, W, f} V \to Y$, $(w, y, v) \mapsto y'$ is an isomorphism.
Thus if we can show that $X$ is an algebraic space, then we are done. Since $V \to W$ is separated and étale it is representable by Morphisms of Spaces, Lemma 67.51.1 (and Morphisms of Spaces, Lemma 67.39.5). Of course $W^0 \to W$ is representable and étale as it is an open immersion. Thus
\[ W^0 \amalg Y = X \times _{g, W} W^0 \amalg X \times _{g, W, f} V = X \times _{g, W} (W^0 \amalg V) \longrightarrow X \]
is representable, surjective, and étale by Spaces, Lemmas 65.3.3 and 65.5.5. Thus $X$ is an algebraic space by Spaces, Lemma 65.11.2.
$\square$
Lemma 70.19.2. Notation and assumptions as in Lemma 70.19.1. Let $g : X \to W$ correspond to $h : Y \to V$ via the equivalence. Then $g$ is quasi-compact, quasi-separated, separated, locally of finite presentation, of finite presentation, locally of finite type, of finite type, proper, integral, finite, and add more here if and only if $h$ is so.
Proof.
If $g$ is quasi-compact, quasi-separated, separated, locally of finite presentation, of finite presentation, locally of finite type, of finite type, proper, finite, so is $h$ as a base change of $g$ by Morphisms of Spaces, Lemmas 67.8.4, 67.4.4, 67.28.3, 67.23.3, 67.40.3, 67.45.5. Conversely, let $P$ be a property of morphisms of algebraic spaces which is étale local on the base and which holds for the identity morphism of any algebraic space. Since $\{ W^0 \to W, V \to W\} $ is an étale covering, to prove that $g$ has $P$ it suffices to show that $h$ has $P$. Thus we conclude using Morphisms of Spaces, Lemmas 67.8.8, 67.4.12, 67.28.4, 67.23.4, 67.40.2, 67.45.3.
$\square$
Lemma 70.19.3. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \in |X|$ be a closed point such that $U = X \setminus \{ x\} \to X$ is quasi-compact. With $V = \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}^ h) \setminus \{ \mathfrak m_ x^ h\} $ the base change functor
\[ \left\{ \begin{matrix} f : Y \to X\text{ of finite presentation}
\\ f^{-1}(U) \to U\text{ is an isomorphism}
\end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} g : Y \to \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}^ h)\text{ of finite presentation}
\\ g^{-1}(V) \to V\text{ is an isomorphism}
\end{matrix} \right\} \]
is an equivalence of categories.
Proof.
Let $a : (W, w) \to (X, x)$ be an elementary étale neighbourhood of $x$ with $W$ affine as in Decent Spaces, Lemma 68.11.4. Since $x$ is a closed point of $X$ and $w$ is the unique point of $W$ lying over $x$, we see that $w$ is a closed point of $W$. Since $a$ is étale and identifies residue fields at $x$ and $w$, it follows that $a$ induces an isomorphism $a^{-1}x \to x$ (as closed subspaces of $X$ and $W$). Thus we may apply Lemma 70.19.1 and 70.19.2 to reduce the problem to the case where $X$ is an affine scheme.
Assume $X$ is an affine scheme. Recall that $\mathcal{O}_{X, x}^ h$ is the colimit of $\Gamma (U, \mathcal{O}_ U)$ over affine elementary étale neighbourhoods $(U, u) \to (X, x)$. Recall that the category of these neighbourhoods is cofiltered, see Decent Spaces, Lemma 68.11.6 or More on Morphisms, Lemma 37.35.4. Then $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}^ h) = \mathop{\mathrm{lim}}\nolimits U$ and $V = \mathop{\mathrm{lim}}\nolimits U \setminus \{ u\} $ (Lemma 70.4.1) where the limits are taken over the same category. Thus by Lemma 70.7.1 The category on the right is the colimit of the categories for the pairs $(U, u)$. And by the material in the first paragraph, each of these categories is equivalent to the category for the pair $(X, x)$. This finishes the proof.
$\square$
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