The Stacks project

Remark 13.26.15. As promised in Remark 13.21.4 we discuss the connection of the lemma above with the constructions using Cartan-Eilenberg resolutions. Namely, let $T : \mathcal{A} \to \mathcal{B}$ be a left exact functor of abelian categories, assume $\mathcal{A}$ has enough injectives, and let $K^\bullet $ be a bounded below complex of $\mathcal{A}$. We give an alternative construction of the spectral sequences ${}'E$ and ${}''E$ of Lemma 13.21.3.

First spectral sequence. Consider the “stupid” filtration on $K^\bullet $ obtained by setting $F^ p(K^\bullet ) = \sigma _{\geq p}(K^\bullet )$, see Homology, Section 12.15. Note that this stupid in the sense that $d(F^ p(K^\bullet )) \subset F^{p + 1}(K^\bullet )$, compare Homology, Lemma 12.24.3. Note that $\text{gr}^ p(K^\bullet ) = K^ p[-p]$ with this filtration. According to Lemma 13.26.14 there is a spectral sequence with $E_1$ term

as in the spectral sequence ${}'E_ r$. Observe moreover that the differentials $E_1^{p, q} \to E_1^{p + 1, q}$ agree with the differentials in $'{}E_1$, see Homology, Lemma 12.24.3 part (2) and the description of ${}'d_1$ in the proof of Lemma 13.21.3.

Second spectral sequence. Consider the filtration on the complex $K^\bullet $ obtained by setting $F^ p(K^\bullet ) = \tau _{\leq -p}(K^\bullet )$, see Homology, Section 12.15. The minus sign is necessary to get a decreasing filtration. Note that $\text{gr}^ p(K^\bullet )$ is quasi-isomorphic to $H^{-p}(K^\bullet )[p]$ with this filtration. According to Lemma 13.26.14 there is a spectral sequence with $E_1$ term

with $i = 2p + q$ and $j = -p$. (This looks unnatural, but note that we could just have well developed the whole theory of filtered complexes using increasing filtrations, with the end result that this then looks natural, but the other one doesn't.) We leave it to the reader to see that the differentials match up.

Actually, given a Cartan-Eilenberg resolution $K^\bullet \to I^{\bullet , \bullet }$ the induced morphism $K^\bullet \to \text{Tot}(I^{\bullet , \bullet })$ into the associated total complex will be a filtered injective resolution for either filtration using suitable filtrations on $\text{Tot}(I^{\bullet , \bullet })$. This can be used to match up the spectral sequences exactly.