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*}

15.64 Spectral sequences for Ext

In this section we collect various spectral sequences that come up when considering the Ext functors. For any pair of objects $L$, $K$ of the derived category $D(R)$ of a ring $R$ we denote

\[ \mathop{\mathrm{Ext}}\nolimits ^ n_ R(L, K) = \mathop{\mathrm{Hom}}\nolimits _{D(R)}(L, K[n]) \]

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

For $M$ an $R$-module and $K \in D^+(R)$ there is a spectral sequence

15.64.0.1
\begin{equation} \label{more-algebra-equation-first-ss-ext} \mathop{\mathrm{Ext}}\nolimits _ R^ j(M, H^ i(K)) \Rightarrow \text{Ext}_ R^{i + j}(M, K) \end{equation}

and if $K$ is represented by the bounded below complex $K^\bullet $ of $R$-modules there is a spectral sequence

15.64.0.2
\begin{equation} \label{more-algebra-equation-second-ss-ext} \mathop{\mathrm{Ext}}\nolimits _ R^ j(M, K^ i) \Rightarrow \text{Ext}_ R^{i + j}(M, K) \end{equation}

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