Lemma 50.18.5. Consider a commutative diagram
\[ \xymatrix{ Y' \ar[d]_{f'} \ar[r]_ b & Y \ar[d]^ f \\ X' \ar[r]^ a & X } \]
of schemes which induces an isomorphism of $Y'$ with an open subscheme of $X' \times _ X Y$. Assume $f$ is locally quasi-finite and syntomic. Then for every $y' \in Y'$ we can find opens $V' \subset Y'$, $V \subset Y$, $U' \subset X$, $U \subset X$ with $y' \in V'$, with $f'(V') \subset U'$, $b(V') \subset V$, $a(U') \subset U$, $f(V) \subset U$, and such that for $p \geq 0$ there are maps $c^ p_{V/U}$ and $c^ p_{V'/U'}$ satisfying (1) and (2) which are compatible with the diagram
\[ \xymatrix{ V' \ar[d] \ar[r]_ b & V \ar[d] \\ U' \ar[r] & U } \]
in the sense explained in Lemma 50.18.4.
Proof.
It is clear that we may replace $Y'$ by $X' \times _ X Y$ in order to prove this. Pick affine opens $V \subset Y$ and $U \subset X$ with $V$ containing the image of $y'$ as in Discriminants, Lemma 49.10.1 part (5). Pick an affine open $U' \subset X'$ containing the image of $y'$ and mapping into $U$. After replacing $X, Y, X', Y'$ by $U, V, U', U' \times _ U V$ we may assume we are in the situation described in the next paragraph.
Assume that $X' = \mathop{\mathrm{Spec}}(A')$, $X = \mathop{\mathrm{Spec}}(A)$, $Y = \mathop{\mathrm{Spec}}(B)$, and $Y' = \mathop{\mathrm{Spec}}(A' \otimes _ A B)$ with $B = A[x_1, \ldots , x_ n]/(f_1, \ldots , f_ n)$ a relative global complete intersection over $A$. Then also $B' = A'[x_1, \ldots , x_ n]/(f'_1, \ldots , f'_ n)$ is a relative global complete intersection over $A'$ where $f'_ i$ is the image of $f_ i$ in $A'[x_1, \ldots , x_ n]$, see Algebra, Lemma 10.136.9. The construction in Remark 50.18.3 provides us with the maps $c^ p_{V/U}$ and $c^ p_{V'/U'}$. These maps are compatible by Lemma 50.18.2 and the compatibility of the presentations of $B$ and $B'$ over $A$ and $A'$. Some details omitted.
$\square$
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