Lemma 62.4.3. Let $R'/R$ be an extension of discrete valuation rings inducing fraction field extension $K'/K$ and residue field extension $\kappa '/\kappa $ (More on Algebra, Definition 15.111.1). Let $X$ be locally of finite type over $R$. Denote $X' = X_{R'}$. Then the diagram
\[ \xymatrix{ Z_ r(X'_{K'}) \ar[rr]_{sp_{X'/R'}} & & Z_ r(X'_{\kappa '}) \\ Z_ r(X_ K) \ar[rr]^{sp_{X/R}} \ar[u] & & Z_ r(X_\kappa ) \ar[u] } \]
commutes where $r \geq 0$ and the vertical arrows are base change maps.
Proof.
Observe that $X'_{K'} = X_{K'} = X_ K \times _{\mathop{\mathrm{Spec}}(K)} \mathop{\mathrm{Spec}}(K')$ and similarly for closed fibres, so that the vertical arrows indeed make sense (see Section 62.3). Now if $Z \subset X_ K$ is an integral closed subscheme with scheme theoretic image $\overline{Z} \subset X$, then we see that $Z_{K'} \subset X_{K'}$ is a closed subscheme with scheme theoretic image $\overline{Z}_{R'} \subset X_{R'}$. The base change of $[Z]$ is $[Z_{K'}]_ r = [\overline{Z}_{K'}]_ r$ by definition. We have
\[ sp_{X/R}([Z]) = [\overline{Z}_\kappa ]_ r \quad \text{and}\quad sp_{X'/R'}([\overline{Z}_{K'}]_ r) = [(\overline{Z}_{R'})_{\kappa '}]_ r \]
by Lemma 62.4.1. Since $(\overline{Z}_{R'})_{\kappa '} = (\overline{Z}_\kappa )_{\kappa '}$ we conclude.
$\square$
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