Example 21.31.1. 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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