Lemma 13.26.14. Let $\mathcal{A}, \mathcal{B}$ be abelian categories. Let $T : \mathcal{A} \to \mathcal{B}$ be a left exact functor. Assume $\mathcal{A}$ has enough injectives. Let $(K^\bullet , F)$ be an object of $\text{Comp}^{+}(\text{Fil}^ f(\mathcal{A}))$. There exists a spectral sequence $(E_ r, d_ r)_{r\geq 0}$ consisting of bigraded objects $E_ r$ of $\mathcal{B}$ and $d_ r$ of bidegree $(r, - r + 1)$ and with
\[ E_1^{p, q} = R^{p + q}T(\text{gr}^ p(K^\bullet )) \]
Moreover, this spectral sequence is bounded, converges to $R^*T(K^\bullet )$, and induces a finite filtration on each $R^ nT(K^\bullet )$. The construction of this spectral sequence is functorial in the object $K^\bullet $ of $\text{Comp}^{+}(\text{Fil}^ f(\mathcal{A}))$ and the terms $(E_ r, d_ r)$ for $r \geq 1$ do not depend on any choices.
Proof.
Choose a filtered quasi-isomorphism $K^\bullet \to I^\bullet $ with $I^\bullet $ a bounded below complex of filtered injective objects, see Lemma 13.26.9. Consider the complex $RT(K^\bullet ) = T_{ext}(I^\bullet )$, see (13.26.13.6). Thus we can consider the spectral sequence $(E_ r, d_ r)_{r \geq 0}$ associated to this as a filtered complex in $\mathcal{B}$, see Homology, Section 12.24. By Homology, Lemma 12.24.2 we have $E_1^{p, q} = H^{p + q}(\text{gr}^ p(T(I^\bullet )))$. By Equation (13.26.13.3) we have $E_1^{p, q} = H^{p + q}(T(\text{gr}^ p(I^\bullet )))$, and by definition of a filtered injective resolution the map $\text{gr}^ p(K^\bullet ) \to \text{gr}^ p(I^\bullet )$ is an injective resolution. Hence $E_1^{p, q} = R^{p + q}T(\text{gr}^ p(K^\bullet ))$.
On the other hand, each $I^ n$ has a finite filtration and hence each $T(I^ n)$ has a finite filtration. Thus we may apply Homology, Lemma 12.24.11 to conclude that the spectral sequence is bounded, converges to $H^ n(T(I^\bullet )) = R^ nT(K^\bullet )$ moreover inducing finite filtrations on each of the terms.
Suppose that $K^\bullet \to L^\bullet $ is a morphism of $\text{Comp}^{+}(\text{Fil}^ f(\mathcal{A}))$. Choose a filtered quasi-isomorphism $L^\bullet \to J^\bullet $ with $J^\bullet $ a bounded below complex of filtered injective objects, see Lemma 13.26.9. By our results above, for example Lemma 13.26.11, there exists a diagram
\[ \xymatrix{ K^\bullet \ar[r] \ar[d] & L^\bullet \ar[d] \\ I^\bullet \ar[r] & J^\bullet } \]
which commutes up to homotopy. Hence we get a morphism of filtered complexes $T(I^\bullet ) \to T(J^\bullet )$ which gives rise to the morphism of spectral sequences, see Homology, Lemma 12.24.4. The last statement follows from this.
$\square$
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