The Stacks project

\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*}

Lemma 21.26.3. In the situation above assume

  1. $h_ X^\# = h_ Y^\# \amalg _{h_ E^\# } h_ Z^\# $, and

  2. $h_ E^\# \to h_ Y^\# $ is injective.

Then the construction of Lemma 21.26.1 produces a distinguished triangle

\[ R\Gamma (X, K) \to R\Gamma (Z, K) \oplus R\Gamma (Y, K) \to R\Gamma (E, K) \to R\Gamma (X, K)[1] \]

functorial for $K$ in $D(\mathcal{C})$.

Proof. We can represent $K$ by a K-injective complex whose terms are injective abelian sheaves, see Section 21.20. Thus it suffices to show: if $\mathcal{I}$ is an injective abelian sheaf, then

\[ 0 \to \mathcal{I}(X) \to \mathcal{I}(Z) \oplus \mathcal{I}(Y) \to \mathcal{I}(E) \to 0 \]

is a short exact sequence. The first arrow is injective because by condition (1) the map $h_ Y \amalg h_ Z \to h_ X$ becomes surjective after sheafification, which means that $\{ Y \to X, Z \to X\} $ can be refined by a covering of $X$. The last arrow is surjective because $\mathcal{I}(Y) \to \mathcal{I}(E)$ is surjective. Namely, we have $\mathcal{I}(E) = \mathop{\mathrm{Hom}}\nolimits (\mathbf{Z}_ E^\# , \mathcal{I})$, $\mathcal{I}(Y) = \mathop{\mathrm{Hom}}\nolimits (\mathbf{Z}_ Y^\# , \mathcal{I})$, the map $\mathbf{Z}_ E^\# \to \mathbf{Z}_ Y^\# $ is injective by (2), and $\mathcal{I}$ is an injective abelian sheaf. Please compare with Modules on Sites, Section 18.5. Finally, suppose we have $s \in \mathcal{I}(Y)$ and $t \in \mathcal{F}(Z)$ mapping to the same element of $\mathcal{I}(E)$. Then $s$ and $t$ define a map

\[ s \amalg t : h_ Y^\# \amalg h_ Z^\# \longrightarrow \mathcal{I} \]

which by assumption factors through $h_ Y^\# \amalg _{h_ E^\# } h_ Z^\# $. Thus by assumption (1) we obtain a unique map $h_ X^\# \to \mathcal{I}$ which corresponds to an element of $\mathcal{I}(X)$ restricting to $s$ on $Y$ and $t$ on $Z$. $\square$


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