The Stacks project

Lemma 20.39.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $X = U \cup V$ be the union of two open subspaces of $X$. Suppose given

  1. an object $A$ of $D(\mathcal{O}_ U)$,

  2. an object $B$ of $D(\mathcal{O}_ V)$, and

  3. an isomorphism $c : A|_{U \cap V} \to B|_{U \cap V}$.

Then there exists an object $F$ of $D(\mathcal{O}_ X)$ and isomorphisms $f : F|_ U \to A$, $g : F|_ V \to B$ such that $c = g|_{U \cap V} \circ f^{-1}|_{U \cap V}$. Moreover, given

  1. an object $E$ of $D(\mathcal{O}_ X)$,

  2. a morphism $a : A \to E|_ U$ of $D(\mathcal{O}_ U)$,

  3. a morphism $b : B \to E|_ V$ of $D(\mathcal{O}_ V)$,

such that

\[ a|_{U \cap V} = b|_{U \cap V} \circ c. \]

Then there exists a morphism $F \to E$ in $D(\mathcal{O}_ X)$ whose restriction to $U$ is $a \circ f$ and whose restriction to $V$ is $b \circ g$.

Proof. Denote $j_ U$, $j_ V$, $j_{U \cap V}$ the corresponding open immersions. Choose a distinguished triangle

\[ F \to Rj_{U, *}A \oplus Rj_{V, *}B \to Rj_{U \cap V, *}(B|_{U \cap V}) \to F[1] \]

where the map $Rj_{V, *}B \to Rj_{U \cap V, *}(B|_{U \cap V})$ is the obvious one and where $Rj_{U, *}A \to Rj_{U \cap V, *}(B|_{U \cap V})$ is the composition of $Rj_{U, *}A \to Rj_{U \cap V, *}(A|_{U \cap V})$ with $Rj_{U \cap V, *}c$. Restricting to $U$ we obtain

\[ F|_ U \to A \oplus (Rj_{V, *}B)|_ U \to (Rj_{U \cap V, *}(B|_{U \cap V}))|_ U \to F|_ U[1] \]

Denote $j : U \cap V \to U$. Compatibility of restriction to opens and cohomology shows that both $(Rj_{V, *}B)|_ U$ and $(Rj_{U \cap V, *}(B|_{U \cap V}))|_ U$ are canonically isomorphic to $Rj_*(B|_{U \cap V})$. Hence the second arrow of the last displayed diagram has a section, and we conclude that the morphism $F|_ U \to A$ is an isomorphism. Similarly, the morphism $F|_ V \to B$ is an isomorphism. The existence of the morphism $F \to E$ follows from the Mayer-Vietoris sequence for $\mathop{\mathrm{Hom}}\nolimits $, see Lemma 20.31.3. $\square$


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