Lemma 86.9.3. Let $S$ be a scheme. Let $X \to Y$ be a proper, flat morphism of algebraic spaces which is of finite presentation. If $(\omega _ j^\bullet , \tau _ j)$, $j = 1, 2$ are two relative dualizing complexes on $X/Y$, then there is a unique isomorphism $(\omega _1^\bullet , \tau _1) \to (\omega _2^\bullet , \tau _2)$.
Proof. Consider $g : Y' \to Y$ étale with $Y'$ an affine scheme and denote $X' = Y' \times _ Y X$ the base change. By Definition 86.9.1 and the discussion following, there is a unique isomorphism $\iota : (\omega _1^\bullet |_{X'}, \tau _1|_{Y'}) \to (\omega _2^\bullet |_{X'}, \tau _2|_{Y'})$. If $Y'' \to Y'$ is a further étale morphism of affines and $X'' = Y'' \times _ Y X$, then $\iota |_{X''}$ is the unique isomorphism $(\omega _1^\bullet |_{X''}, \tau _1|_{Y''}) \to (\omega _2^\bullet |_{X''}, \tau _2|_{Y''})$ (by uniqueness). Also we have
\[ \text{Ext}^ p_{X'}(\omega _1^\bullet |_{X'}, \omega _2^\bullet |_{X'}) = 0, \quad p < 0 \]
because $\mathcal{O}_{X'} \cong R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{X'}}(\omega _1^\bullet |_{X'}, \omega _1^\bullet |_{X'}) \cong R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{X'}}(\omega _1^\bullet |_{X'}, \omega _2^\bullet |_{X'})$ by Lemma 86.9.2.
Choose a étale hypercovering $b : V \to Y$ such that each $V_ n = \coprod _{i \in I_ n} Y_{n, i}$ with $Y_{n, i}$ affine. This is possible by Hypercoverings, Lemma 25.12.2 and Remark 25.12.9 (to replace the hypercovering produced in the lemma by the one having disjoint unions in each degree). Denote $X_{n, i} = Y_{n, i} \times _ Y X$ and $U_ n = V_ n \times _ Y X$ so that we obtain an étale hypercovering $a : U \to X$ (Hypercoverings, Lemma 25.12.4) with $U_ n = \coprod X_{n, i}$. The assumptions of Simplicial Spaces, Lemma 85.35.1 are satisfied for $a : U \to X$ and the complexes $\omega _1^\bullet $ and $\omega _2^\bullet $. Hence we obtain a unique morphism $\iota : \omega _1^\bullet \to \omega _2^\bullet $ whose restriction to $X_{0, i}$ is the unique isomorphism $(\omega _1^\bullet |_{X_{0, i}}, \tau _1|_{Y_{0, i}}) \to (\omega _2^\bullet |_{X_{0, i}}, \tau _2|_{Y_{0, i}})$ We still have to see that the diagram
\[ \xymatrix{ Rf_*\omega _1^\bullet \ar[rd]_{\tau _1} \ar[rr]_{Rf_*\iota } & & Rf_*\omega _1^\bullet \ar[ld]^{\tau _2} \\ & \mathcal{O}_ Y } \]
is commutative. However, we know that $Rf_*\omega _1^\bullet $ and $Rf_*\omega _2^\bullet $ have vanishing cohomology sheaves in positive degrees (Lemma 86.9.2) thus this commutativity may be proved after restricting to the affines $Y_{0, i}$ where it holds by construction. $\square$
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