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The Stacks project

Lemma 49.10.7. Let f : Y \to X be a morphism of schemes. If f satisfies the equivalent conditions of Lemma 49.10.1 then for every y \in Y there exist n, d and a commutative diagram

\xymatrix{ Y \ar[d] & V \ar[d] \ar[l] \ar[r] & Y_{n, d} \ar[d] \\ X & U \ar[l] \ar[r] & X_{n, d} }

where U \subset X and V \subset Y are open, where Y_{n, d} \to X_{n, d} is as in Example 49.10.5, and where the square on the right hand side is cartesian.

Proof. By Lemma 49.10.1 we can choose U and V affine so that U = \mathop{\mathrm{Spec}}(R) and V = \mathop{\mathrm{Spec}}(S) with S = R[y_1, \ldots , y_ n]/(g_1, \ldots , g_ n). With notation as in Example 49.10.5 if we pick d large enough, then we can write each g_ i as g_ i = \sum _{E \in T} g_{i, E}y^ E with g_{i, E} \in R. Then the map A \to R sending a_{i, E} to g_{i, E} and the map B \to S sending x_ i \to y_ i give a cocartesian diagram of rings

\xymatrix{ S & B \ar[l] \\ R \ar[u] & A \ar[l] \ar[u] }

which proves the lemma. \square


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