The Stacks project

Lemma 75.10.4. Let $S$ be a scheme. Let $(U \subset X, V \to X)$ be an elementary distinguished square of algebraic spaces over $S$. For objects $E$, $F$ of $D(\mathcal{O}_ X)$ we have a Mayer-Vietoris sequence

\[ \xymatrix{ & \ldots \ar[r] & \mathop{\mathrm{Ext}}\nolimits ^{-1}(E_{U \times _ X V}, F_{U \times _ X V}) \ar[lld] \\ \mathop{\mathrm{Hom}}\nolimits (E, F) \ar[r] & \mathop{\mathrm{Hom}}\nolimits (E_ U, F_ U) \oplus \mathop{\mathrm{Hom}}\nolimits (E_ V, F_ V) \ar[r] & \mathop{\mathrm{Hom}}\nolimits (E_{U \times _ X V}, F_{U \times _ X V}) } \]

where the subscripts denote restrictions to the relevant opens and the $\mathop{\mathrm{Hom}}\nolimits $'s are taken in the relevant derived categories.

Proof. Use the distinguished triangle of Lemma 75.10.1 to obtain a long exact sequence of $\mathop{\mathrm{Hom}}\nolimits $'s (from Derived Categories, Lemma 13.4.2) and use that $\mathop{\mathrm{Hom}}\nolimits (j_{U!}E|_ U, F) = \mathop{\mathrm{Hom}}\nolimits (E|_ U, F|_ U)$ by Cohomology on Sites, Lemma 21.20.8. $\square$

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