The Stacks project

Lemma 105.16.5. Let $a : \mathcal{Y} \to \mathcal{X}$ and $b : \mathcal{Z} \to \mathcal{X}$ be representable by schemes, quasi-compact, quasi-separated, and flat. Then $a_{\mathit{QCoh}, *}\mathcal{O}_\mathcal {Y} \otimes _{\mathcal{O}_\mathcal {X}} b_{\mathit{QCoh}, *}\mathcal{O}_\mathcal {Z} = f_{\mathit{QCoh}, *}\mathcal{O}_{\mathcal{Y} \times _\mathcal {X} \mathcal{Z}}$ where $f : \mathcal{Y} \times _\mathcal {X} \mathcal{Z} \to \mathcal{X}$ is the obvious morphism.

Proof. We abbreviate $\mathcal{P} = \mathcal{Y} \times _\mathcal {X} \mathcal{Z}$. Since $a \circ \text{pr}_1 = f$ and $b \circ \text{pr}_2 = f$ we obtain maps $a_*\mathcal{O}_\mathcal {Y} \to f_*\mathcal{O}_\mathcal {P}$ and $b_*\mathcal{O}_\mathcal {Z} \to f_*\mathcal{O}_\mathcal {P}$ (using relative pullback maps, see Sites, Section 7.45). Hence we obtain a relative cup product

\[ \mu : a_*\mathcal{O}_\mathcal {Y} \otimes _{\mathcal{O}_\mathcal {X}} b_*\mathcal{O}_\mathcal {Z} \longrightarrow f_*\mathcal{O}_{\mathcal{Y} \times _\mathcal {X} \mathcal{Z}} \]

Applying $Q$ and its compatibility with tensor products (Cohomology of Stacks, Remark 102.10.6) we obtain an arrow $Q(\mu ) : a_{\mathit{QCoh}, *}\mathcal{O}_\mathcal {Y} \otimes _{\mathcal{O}_\mathcal {X}} b_{\mathit{QCoh}, *}\mathcal{O}_\mathcal {Z} \to f_{\mathit{QCoh}, *}\mathcal{O}_{\mathcal{Y} \times _\mathcal {X} \mathcal{Z}}$ in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$. Next, choose a scheme $U$ and a surjective smooth morphism $U \to \mathcal{X}$. It suffices to prove the restriction of $Q(\mu )$ to $U_{\acute{e}tale}$ is an isomorphism, see Cohomology of Stacks, Section 102.12. In turn, by the material in the same section, it suffices to prove the restriction of $\mu $ to $U_{\acute{e}tale}$ is an isomorphism (this uses that the source and target of $\mu $ are locally quasi-coherent modules with the base change property). Moreover, we may compute pushforwards in the étale topology, see Cohomology of Stacks, Proposition 102.8.1. Then finally, we see that $a_*\mathcal{O}_\mathcal {Y}|_{U_{\acute{e}tale}} = (V \to U)_{small, *}\mathcal{O}_ V$ where $V = U \times _\mathcal {X} \mathcal{Y}$. Similarly for $b_*$ and $f_*$. Thus the result follows from the Künneth formula for flat, quasi-compact, quasi-separated morphisms of schemes, see Derived Categories of Schemes, Lemma 36.23.1. $\square$

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