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*}

21.32 Spectral sequences for Ext

In this section we collect various spectral sequences that come up when considering the Ext functors. For any pair of complexes $\mathcal{G}^\bullet , \mathcal{F}^\bullet $ of complexes of modules on a ringed site $(\mathcal{C}, \mathcal{O})$ we denote

\[ \mathop{\mathrm{Ext}}\nolimits ^ n_\mathcal {O}(\mathcal{G}^\bullet , \mathcal{F}^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O})}(\mathcal{G}^\bullet , \mathcal{F}^\bullet [n]) \]

according to our general conventions in Derived Categories, Section 13.27.

Example 21.32.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{K}^\bullet $ be a bounded above complex of $\mathcal{O}$-modules. Let $\mathcal{F}$ be an $\mathcal{O}$-module. Then there is a spectral sequence with $E_2$-page

\[ E_2^{i, j} = \mathop{\mathrm{Ext}}\nolimits _\mathcal {O}^ i(H^{-j}(\mathcal{K}^\bullet ), \mathcal{F}) \Rightarrow \mathop{\mathrm{Ext}}\nolimits _\mathcal {O}^{i + j}(\mathcal{K}^\bullet , \mathcal{F}) \]

and another spectral sequence with $E_1$-page

\[ E_1^{i, j} = \mathop{\mathrm{Ext}}\nolimits _\mathcal {O}^ j(\mathcal{K}^{-i}, \mathcal{F}) \Rightarrow \mathop{\mathrm{Ext}}\nolimits _\mathcal {O}^{i + j}(\mathcal{K}^\bullet , \mathcal{F}). \]

To construct these spectral sequences choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet $ and consider the two spectral sequences coming from the double complex $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{K}^\bullet , \mathcal{I}^\bullet )$, see Homology, Section 12.22.


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