Remark 19.13.10. Let $\mathcal{A}$ be a Grothendieck abelian category. Let $K \in D(\mathcal{A})$. Let $M^\bullet$ be a filtered complex of $\mathcal{A}$, see Homology, Definition 12.24.1. For ease of notation denote $M$, $M/F^ pM$, $\text{gr}^ pM$ the object of $D(\mathcal{A})$ represented by $M^\bullet$, $M^\bullet /F^ pM^\bullet$, $\text{gr}^ pM^\bullet$. Dually to Remark 19.13.8 we can construct a spectral sequence $(E_ r, d_ r)_{r \geq 1}$ of bigraded objects of $\mathcal{A}$ with $d_ r$ of bidgree $(r, -r + 1)$ and with

$E_1^{p, q} = \mathop{\mathrm{Ext}}\nolimits ^{p + q}(\text{gr}^{-p}M, K)$

If for every $n$ we have

$\mathop{\mathrm{Ext}}\nolimits ^ n(M/F^ pM, K) = 0 \text{ for } p \ll 0 \quad \text{and}\quad \mathop{\mathrm{Ext}}\nolimits ^ n(M/F^ pM, K) = \mathop{\mathrm{Ext}}\nolimits ^ n(M, K) \text{ for } p \gg 0$

then the spectral sequence is bounded and converges to $\mathop{\mathrm{Ext}}\nolimits ^{p + q}(M, K)$. Namely, choose a K-injective complex $I^\bullet$ with injective terms representing $K$, see Theorem 19.12.6. Consider the complex

$\mathop{\mathrm{Hom}}\nolimits ^\bullet (M^\bullet , I^\bullet )$

defined exactly as in More on Algebra, Section 15.71. Setting

$F^ p\mathop{\mathrm{Hom}}\nolimits ^\bullet (M^\bullet , I^\bullet ) = \mathop{\mathrm{Hom}}\nolimits ^\bullet (M^\bullet /F^{-p + 1}M^\bullet , I^\bullet )$

we obtain a filtered complex (note sign and shift in filtration). The spectral sequence of Homology, Section 12.24 has differentials and terms as described above; details omitted. The boundedness and convergence follows from Homology, Lemma 12.24.13.

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