The Stacks project

Remark 19.13.8. Let $\mathcal{A}$ be a Grothendieck abelian category. Let $K^\bullet $ be a filtered complex of $\mathcal{A}$, see Homology, Definition 12.24.1. For ease of notation denote $K$, $F^ pK$, $\text{gr}^ pK$ the object of $D(\mathcal{A})$ represented by $K^\bullet $, $F^ pK^\bullet $, $\text{gr}^ pK^\bullet $. Let $M \in D(\mathcal{A})$. Using Lemma 19.13.7 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

\[ E_1^{p, q} = \mathop{\mathrm{Ext}}\nolimits ^{p + q}(M, \text{gr}^ pK) \]

If for every $n$ we have

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

then the spectral sequence is bounded and converges to $\mathop{\mathrm{Ext}}\nolimits ^{p + q}(M, K)$. Namely, choose any complex $M^\bullet $ representing $M$, choose $j : K^\bullet \to J^\bullet $ as in the lemma, and consider the complex

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

defined exactly as in More on Algebra, Section 15.71. Setting $F^ p\mathop{\mathrm{Hom}}\nolimits ^\bullet (M^\bullet , I^\bullet ) = \mathop{\mathrm{Hom}}\nolimits ^\bullet (M^\bullet , F^ pI^\bullet )$ we obtain a filtered complex. 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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