## 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 12.25.

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