\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

The Stacks project

Lemma 67.10.5. Let $S$ be a scheme. Let $(U \subset X, V \to X)$ be an elementary distinguished square of algebraic spaces over $S$. For an object $E$ of $D(\mathcal{O}_ X)$ we have a distinguished triangle

\[ R\Gamma (X, E) \to R\Gamma (U, E) \oplus R\Gamma (V, E) \to R\Gamma (U \times _ X V, E) \to R\Gamma (X, E)[1] \]

and in particular a long exact cohomology sequence

\[ \ldots \to H^ n(X, E) \to H^ n(U, E) \oplus H^ n(V, E) \to H^ n(U \times _ X V, E) \to H^{n + 1}(X, E) \to \ldots \]

The construction of the distinguished triangle and the long exact sequence is functorial in $E$.

Proof. Choose a K-injective complex $\mathcal{I}^\bullet $ representing $E$ whose terms $\mathcal{I}^ n$ are injective objects of $\textit{Mod}(\mathcal{O}_ X)$, see Injectives, Theorem 19.12.6. In the proof of Lemma 67.10.2 we found a short exact sequence of complexes

\[ 0 \to \mathcal{I}^\bullet \to j_{U, *}\mathcal{I}^\bullet |_ U \oplus j_{V, *}\mathcal{I}^\bullet |_ V \to j_{U \times _ X V, *}\mathcal{I}^\bullet |_{U \times _ X V} \to 0 \]

Since $H^1(X, \mathcal{I}^ n) = 0$, we see that taking global sections gives an exact sequence of complexes

\[ 0 \to \Gamma (X, \mathcal{I}^\bullet ) \to \Gamma (U, \mathcal{I}^\bullet ) \oplus \Gamma (V, \mathcal{I}^\bullet ) \to \Gamma (U \times _ X V, \mathcal{I}^\bullet ) \to 0 \]

Since these complexes represent $R\Gamma (X, E)$, $R\Gamma (U, E)$, $R\Gamma (V, E)$, and $R\Gamma (U \times _ X V, E)$ we get a distinguished triangle by Derived Categories, Section 13.12 and especially Lemma 13.12.1. $\square$


Comments (2)

Comment #2792 by on

There shouldn't be a 0 in the long exact cohomology sequence, rather an .


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