The Stacks project

Remark 48.6.1. Consider a cartesian diagram

\[ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y } \]

of quasi-compact and quasi-separated schemes with $(g, f)$ Tor independent. Let $V \subset Y$ and $V' \subset Y'$ be affine opens with $g(V') \subset V$. Form the cartesian diagrams

\[ \vcenter { \xymatrix{ U \ar[r] \ar[d] & X \ar[d] \\ V \ar[r] & Y } } \quad \text{and}\quad \vcenter { \xymatrix{ U' \ar[r] \ar[d] & X' \ar[d] \\ V' \ar[r] & Y' } } \]

Assume (48.4.1.1) with respect to $K$ and the first diagram and (48.4.1.1) with respect to $Lg^*K$ and the second diagram are isomorphisms. Then the restriction of the base change map (48.5.0.1)

\[ L(g')^*a(K) \longrightarrow a'(Lg^*K) \]

to $U'$ is isomorphic to the base change map (48.5.0.1) for $K|_ V$ and the cartesian diagram

\[ \xymatrix{ U' \ar[r] \ar[d] & U \ar[d] \\ V' \ar[r] & V } \]

This follows from the fact that (48.4.1.1) is a special case of the base change map (48.5.0.1) and that the base change maps compose correctly if we stack squares horizontally, see Lemma 48.5.2. Thus in order to check the base change map restricted to $U'$ is an isomorphism it suffices to work with the last diagram.

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