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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