Lemma 20.33.3 (08BW)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 20: Cohomology of Sheaves
Section 20.33: Unbounded Mayer-Vietoris
Lemma 20.33.3 (cite)

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Lemma 20.33.3. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $X = U \cup V$ be the union of two open subspaces of $X$. For objects $E$, $F$ of $D(\mathcal{O}_ X)$ we have a Mayer-Vietoris sequence

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

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

Proof. Use the distinguished triangle of Lemma 20.33.1 to obtain a long exact sequence of $\mathop{\mathrm{Hom}}\nolimits $'s (from Derived Categories, Lemma 13.4.2) and use that

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(j_{U!}E|_ U, F) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(E|_ U, F|_ U) \]

by Lemma 20.32.8. $\square$

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