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

If for every $n$ we have

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

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

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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