Lemma 32.22.9. In Situation 32.22.1 suppose that we have a cartesian diagram
\xymatrix{ X^1 \ar[r]_ p \ar[d]_ q & X^3 \ar[d]^ a \\ X^2 \ar[r]^ b & X^4 }
of schemes quasi-separated and of finite type over S. For each j = 1, 2, 3, 4 choose i_ j \in I and a diagram
\xymatrix{ X^ j \ar[r] \ar[d] & W^ j \ar[d] \\ S \ar[r] & S_{i_ j} }
as in (32.22.2.1). Let X^ j = \mathop{\mathrm{lim}}\nolimits _{i \geq i_ j} X^ j_ i be the corresponding limit descriptions as in Lemma 32.22.4. Let (a_ i)_{i \geq i_5}, (b_ i)_{i \geq i_6}, (p_ i)_{i \geq i_7}, and (q_ i)_{i \geq i_8} be the corresponding morphisms of systems constructed in Lemma 32.22.4. Then there exists an i_9 \geq \max (i_5, i_6, i_7, i_8) such that for i \geq i_9 we have a_ i \circ p_ i = b_ i \circ q_ i and such that
(q_ i, p_ i) : X^1_ i \longrightarrow X^2_ i \times _{b_ i, X^4_ i, a_ i} X^3_ i
is a closed immersion. If a and b are flat and of finite presentation, then there exists an i_{10} \geq \max (i_5, i_6, i_7, i_8, i_9) such that for i \geq i_{10} the last displayed morphism is an isomorphism.
Proof.
According to the discussion in Remark 32.22.5 the choice of W^1 fitting into a diagram as in (32.22.2.1) is immaterial for the truth of the lemma. Thus we may choose W^1 = W^2 \times _{W^4} W^3. Then it is immediate from the construction of X^1_ i that a_ i \circ p_ i = b_ i \circ q_ i and that
(q_ i, p_ i) : X^1_ i \longrightarrow X^2_ i \times _{b_ i, X^4_ i, a_ i} X^3_ i
is a closed immersion.
If a and b are flat and of finite presentation, then so are p and q as base changes of a and b. Thus we can apply Lemma 32.22.6 to each of a, b, p, q, and a \circ p = b \circ q. It follows that there exists an i_9 \in I such that
(q_ i, p_ i) : X^1_ i \to X^2_ i \times _{X^4_ i} X^3_ i
is the base change of (q_{i_9}, p_{i_9}) by the morphism by the morphism X^4_ i \to X^4_{i_9} for all i \geq i_9. We conclude that (q_ i, p_ i) is an isomorphism for all sufficiently large i by Lemma 32.8.11.
\square
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