Lemma 50.18.4. 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 and assume given maps $c^ p_{Y/X} : \Omega ^ p_{Y/\mathbf{Z}} \to f^*\Omega ^ p_{X/\mathbf{Z}} \otimes _{\mathcal{O}_ Y} \det (\mathop{N\! L}\nolimits _{Y/X})$ for $p \geq 0$ satisfying (1) and (2). Then there is at most one collection of maps $c^ p_{Y'/X'} : \Omega ^ p_{Y'/\mathbf{Z}} \to (f')^*\Omega ^ p_{X'/\mathbf{Z}} \otimes _{\mathcal{O}_{Y'}} \det (\mathop{N\! L}\nolimits _{Y'/X'})$ for $p \geq 0$ satisfying (1) and (2) such that the diagrams
\[ \xymatrix{ b^*\Omega ^ p_{Y/\mathbf{Z}} \ar[rr]_-{b^*c^ p_{Y/X}} \ar[d] & & b^*(f^*\Omega ^ p_{X/\mathbf{Z}} \otimes _{\mathcal{O}_ Y} \det (\mathop{N\! L}\nolimits _{Y/X})) \ar@{=}[r] & (f')^*a^*\Omega ^ p_{X/\mathbf{Z}} \otimes _{\mathcal{O}_ Y} b^*\det (\mathop{N\! L}\nolimits _{Y/X})) \ar[d] \\ \Omega ^ p_{Y'/\mathbf{Z}} \ar[rrr]^-{c^ p_{Y'/X'}} & & & (f')^*\Omega ^ p_{X'/\mathbf{Z}} \otimes _{\mathcal{O}_{Y'}} \det (\mathop{N\! L}\nolimits _{Y'/X'}) } \]
commute for all $p \geq 0$. Here the vertical arrows use the maps $b^*\Omega ^ p_{Y/\mathbf{Z}} \to \Omega ^ p_{Y'/\mathbf{Z}}$ and $a^*\Omega ^ p_{X/\mathbf{Z}} \to \Omega ^ p_{X'/\mathbf{Z}}$ of Section 50.2 as well as the identification $b^*\det (\mathop{N\! L}\nolimits _{Y/X}) = \det (\mathop{N\! L}\nolimits _{Y'/X'})$ of Discriminants, Section 49.13.
Proof.
The map
\[ (f')^*\Omega _{X'/\mathbf{Z}} \oplus b^*\Omega _{Y/\mathbf{Z}} \longrightarrow \Omega _{Y'/\mathbf{Z}} \]
is surjective because $Y'$ is an open subscheme of the fibre product; the corresponding algebra statement is that $\Omega _{B \otimes _ A C/\mathbf{Z}}$ is generated as a module by the images of $\Omega _{B/\mathbf{Z}}$ and $\Omega _{C/\mathbf{Z}}$. Thus for all $p$ the map
\[ \bigoplus \nolimits _{i = 0, \ldots , p} (f')^*\Omega ^ i_{X'/\mathbf{Z}} \otimes b^*\Omega ^{p - i}_{Y/\mathbf{Z}} \longrightarrow \Omega ^ p_{Y'/\mathbf{Z}} \]
is surjective. Conditions (1) and (2) combined with the commutativity of the diagram in the lemma, implies that the map $c^ p_{Y'/X'}$ is determined (but existence is not immediate).
$\square$
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