The Stacks project

Lemma 86.9.6. Let $S$ be a scheme. Consider a cartesian square

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

of algebraic spaces over $S$. Assume $X \to Y$ is proper, flat, and of finite presentation. Let $(\omega _{X/Y}^\bullet , \tau )$ be a relative dualizing complex for $f$. Then $(L(g')^*\omega _{X/Y}^\bullet , Lg^*\tau )$ is a relative dualizing complex for $f'$.

Proof. Observe that $L(g')^*\omega _{X/Y}^\bullet $ is $Y'$-perfect by More on Morphisms of Spaces, Lemma 76.52.6. The other condition of Definition 86.9.1 holds by transitivity of fibre products. $\square$

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