The Stacks project

Lemma 20.31.8. Consider a commutative square

\[ \xymatrix{ (X', \mathcal{O}_{X'}) \ar[r]_{g'} \ar[d]_{f'} & (X, \mathcal{O}_ X) \ar[d]^ f \\ (Y', \mathcal{O}_{Y'}) \ar[r]^ g & (Y, \mathcal{O}_ Y) } \]

of ringed spaces. Let $K, L$ in $D(\mathcal{O}_ X)$. The relative cup product is compatible with the square in the sense that the diagram

\[ \xymatrix{ Lg^*(Rf_*K \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*L) \ar[r] \ar@{=}[d] & Lg^*(Rf_*(K \otimes _{\mathcal{O}_ X}^\mathbf {L} L)) \ar[d] \\ Lg^*Rf_*K \otimes _{\mathcal{O}_{Y'}}^\mathbf {L} Lg^*Rf_*L \ar[d] & R(f')_*L(g')^*(K \otimes _{\mathcal{O}_ X}^\mathbf {L} L) \ar@{=}[d] \\ R(f')_*(L(g')^*K \otimes _{\mathcal{O}_{Y'}} R(f')_*(L(g')^*L \ar[r] & R(f')_*(L(g')^*K \otimes _{\mathcal{O}_{X'}}^\mathbf {L} L(g')^*L) } \]

is commutative in $D(\mathcal{O}_{Y'})$. The horizontal arrows are given by the relative cup product (Remark 20.28.7) and the vertical arrows are given by the base change map (Remark 20.28.3) and Lemma 20.27.3.

Proof. Omitted. $\square$

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