The Stacks project

Remark 20.28.5. Consider a commutative diagram

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

of ringed spaces. Then the base change maps of Remark 20.28.3 for the two squares compose to give the base change map for the outer rectangle. More precisely, the composition

\begin{align*} L(h \circ h')^* \circ Rf_* & = L(h')^* \circ Lh_* \circ Rf_* \\ & \to L(h')^* \circ Rf'_* \circ Lg^* \\ & \to Rf''_* \circ L(g')^* \circ Lg^* \\ & = Rf”_* \circ L(g \circ g')^* \end{align*}

is the base change map for the rectangle. We omit the verification.

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