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 21.29.3. With $\epsilon : (\mathcal{C}_\tau , \mathcal{O}_\tau ) \to (\mathcal{C}_{\tau '}, \mathcal{O}_{\tau '})$ as above. Let

\[ \xymatrix{ E \ar[d] \ar[r] & Y \ar[d] \\ Z \ar[r] & X } \]

be a commutative diagram in the category $\mathcal{C}$ such that

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

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

where ${}^\# $ denotes $\tau $-sheafification. Then for $K' \in D(\mathcal{O}_{\tau '})$ in the essential image of $R\epsilon _*$ the map $c^{K'}_{X, Z, Y, E}$ of Lemma 21.26.1 (using the $\tau '$-topology) is an isomorphism.

Proof. This helper lemma is an almost immediate consequence of Lemma 21.26.3 and we strongly urge the reader skip the proof. Say $K' = R\epsilon _*K$. Choose a K-injective complex of $\mathcal{O}_\tau $-modules $\mathcal{J}^\bullet $ representing $K$. Then $\epsilon _*\mathcal{J}^\bullet $ is a K-injective complex of $\mathcal{O}_{\tau '}$-modules representing $K'$, see Lemma 21.21.9. Next,

\[ 0 \to \mathcal{J}^\bullet (X) \xrightarrow {\alpha } \mathcal{J}^\bullet (Z) \oplus \mathcal{J}^\bullet (Y) \xrightarrow {\beta } \mathcal{J}^\bullet (E) \to 0 \]

is a short exact sequence of complexes of abelian groups, see Lemma 21.26.3 and its proof. Since this is the same as the sequence of complexes of abelian groups which is used to define $c^{K'}_{X, Z, Y, E}$, we conclude. $\square$


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