Lemma 32.22.4. In Situation 32.22.1. Let $f : X \to Y$ be a morphism of schemes quasi-separated and of finite type over $S$. Let
\[ \vcenter { \xymatrix{ X \ar[r] \ar[d] & W \ar[d] \\ S \ar[r] & S_{i_1} } } \quad \text{and}\quad \vcenter { \xymatrix{ Y \ar[r] \ar[d] & V \ar[d] \\ S \ar[r] & S_{i_2} } } \]
be diagrams as in (32.22.2.1). Let $X = \mathop{\mathrm{lim}}\nolimits _{i \geq i_1} X_ i$ and $Y = \mathop{\mathrm{lim}}\nolimits _{i \geq i_2} Y_ i$ be the corresponding limit descriptions as in Lemma 32.22.3. Then there exists an $i_0 \geq \max (i_1, i_2)$ and a morphism
\[ (f_ i)_{i \geq i_0} : (X_ i)_{i \geq i_0} \to (Y_ i)_{i \geq i_0} \]
of inverse systems over $(S_ i)_{i \geq i_0}$ such that such that $f = \mathop{\mathrm{lim}}\nolimits _{i \geq i_0} f_ i$. If $(g_ i)_{i \geq i_0} : (X_ i)_{i \geq i_0} \to (Y_ i)_{i \geq i_0}$ is a second morphism of inverse systems over $(S_ i)_{i \geq i_0}$ such that such that $f = \mathop{\mathrm{lim}}\nolimits _{i \geq i_0} g_ i$ then $f_ i = g_ i$ for all $i \gg i_0$.
Proof.
Since $V \to S_{i_2}$ is of finite presentation and $X = \mathop{\mathrm{lim}}\nolimits _{i \geq i_1} X_ i$ we can appeal to Proposition 32.6.1 to find an $i_0 \geq \max (i_1, i_2)$ and a morphism $h : X_{i_0} \to V$ over $S_{i_2}$ such that $X \to X_{i_0} \to V$ is equal to $X \to Y \to V$. For $i \geq i_0$ we get a commutative solid diagram
\[ \xymatrix{ X \ar[d] \ar[r] & X_ i \ar[r] \ar@{..>}[d] \ar@/_2pc/[dd] |!{[d];[ld]}\hole & X_{i_0} \ar[d]^ h \\ Y \ar[r] \ar[d] & Y_ i \ar[r] \ar[d] & V \ar[d] \\ S \ar[r] & S_ i \ar[r] & S_{i_0} } \]
Since $X \to X_ i$ has scheme theoretically dense image and since $Y_ i$ is the scheme theoretic image of $Y \to S_ i \times _{S_{i_2}} V$ we find that the morphism $X_ i \to S_ i \times _{S_{i_2}} V$ induced by the diagram factors through $Y_ i$ (Morphisms, Lemma 29.6.6). This proves existence.
Uniqueness. Let $E_ i \subset X_ i$ be the equalizer of $f_ i$ and $g_ i$ for $i \geq i_0$. By Schemes, Lemma 26.21.5 $E_ i$ is a locally closed subscheme of $X_ i$. Since $X_ i$ is a closed subscheme of $S_ i \times _{S_{i_0}} X_{i_0}$ and similarly for $Y_ i$ we see that
\[ E_ i = X_ i \times _{(S_ i \times _{S_{i_0}} X_{i_0})} (S_ i \times _{S_{i_0}} E_{i_0}) \]
Thus to finish the proof it suffices to show that $X_ i \to X_{i_0}$ factors through $E_{i_0}$ for some $i \geq i_0$. To do this we will use that $X \to X_{i_0}$ factors through $E_{i_0}$ as both $f_{i_0}$ and $g_{i_0}$ are compatible with $f$. Since $X_ i$ is Noetherian, we see that the underlying topological space $|E_{i_0}|$ is a constructible subset of $|X_{i_0}|$ (Topology, Lemma 5.16.1). Hence $X_ i \to X_{i_0}$ factors through $E_{i_0}$ set theoretically for large enough $i$ by Lemma 32.4.10. For such an $i$ the scheme theoretic inverse image $(X_ i \to X_{i_0})^{-1}(E_{i_0})$ is a closed subscheme of $X_ i$ through which $X$ factors and hence equal to $X_ i$ since $X \to X_ i$ has scheme theoretically dense image by construction. This concludes the proof.
$\square$
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