The Stacks project

Lemma 90.24.2. Let $f : X \to Y$ be a morphism of schemes. Let $U = \mathop{\mathrm{Spec}}(A) \subset X$ and $V = \mathop{\mathrm{Spec}}(B) \subset Y$ be affine opens such that $f(U) \subset V$. There is a canonical map

\[ \widetilde{L_{B/A}} \longrightarrow L_{X/Y}|_ U \]

of complexes which is an isomorphism in $D(\mathcal{O}_ U)$. This map is compatible with restricting to smaller affine opens of $X$ and $Y$.

Proof. By Remark 90.18.5 there is a canonical map of complexes $L_{\mathcal{O}_ X(U)/f^{-1}\mathcal{O}_ Y(U)} \to L_{X/Y}(U)$ of $B = \mathcal{O}_ X(U)$-modules, which is compatible with further restrictions. Using the canonical map $A \to f^{-1}\mathcal{O}_ Y(U)$ we obtain a canonical map $L_{B/A} \to L_{\mathcal{O}_ X(U)/f^{-1}\mathcal{O}_ Y(U)}$ of complexes of $B$-modules. Using the universal property of the $\widetilde{\ }$ functor (see Schemes, Lemma 26.7.1) we obtain a map as in the statement of the lemma. We may check this map is an isomorphism on cohomology sheaves by checking it induces isomorphisms on stalks. This follows immediately from Lemmas 90.18.9 and 90.8.6 (and the description of the stalks of $\mathcal{O}_ X$ and $f^{-1}\mathcal{O}_ Y$ at a point $\mathfrak p \in \mathop{\mathrm{Spec}}(B)$ as $B_\mathfrak p$ and $A_\mathfrak q$ where $\mathfrak q = A \cap \mathfrak p$; references used are Schemes, Lemma 26.5.4 and Sheaves, Lemma 6.21.5). $\square$


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