Lemma 13.21.3. Let $F : \mathcal{A} \to \mathcal{B}$ be a left exact functor of abelian categories. Let $K^\bullet $ be a bounded below complex of $\mathcal{A}$. Let $I^{\bullet , \bullet }$ be a Cartan-Eilenberg resolution for $K^\bullet $. The spectral sequences $({}'E_ r, {}'d_ r)_{r \geq 0}$ and $({}''E_ r, {}''d_ r)_{r \geq 0}$ associated to the double complex $F(I^{\bullet , \bullet })$ satisfy the relations
\[ {}'E_1^{p, q} = R^ qF(K^ p) \quad \text{and} \quad {}''E_2^{p, q} = R^ pF(H^ q(K^\bullet )) \]
Moreover, these spectral sequences are bounded, converge to $H^*(RF(K^\bullet ))$, and the associated induced filtrations on $H^ n(RF(K^\bullet ))$ are finite.
Proof.
We will use the following remarks without further mention:
As $I^{p, \bullet }$ is an injective resolution of $K^ p$ we see that $RF$ is defined at $K^ p[0]$ with value $F(I^{p, \bullet })$.
As $H^ p_ I(I^{\bullet , \bullet })$ is an injective resolution of $H^ p(K^\bullet )$ the derived functor $RF$ is defined at $H^ p(K^\bullet )[0]$ with value $F(H^ p_ I(I^{\bullet , \bullet }))$.
By Homology, Lemma 12.25.4 the total complex $\text{Tot}(I^{\bullet , \bullet })$ is an injective resolution of $K^\bullet $. Hence $RF$ is defined at $K^\bullet $ with value $F(\text{Tot}(I^{\bullet , \bullet }))$.
Consider the two spectral sequences associated to the double complex $L^{\bullet , \bullet } = F(I^{\bullet , \bullet })$, see Homology, Lemma 12.25.1. These are both bounded, converge to $H^*(\text{Tot}(L^{\bullet , \bullet }))$, and induce finite filtrations on $H^ n(\text{Tot}(L^{\bullet , \bullet }))$, see Homology, Lemma 12.25.3. Since $\text{Tot}(L^{\bullet , \bullet }) = \text{Tot}(F(I^{\bullet , \bullet })) = F(\text{Tot}(I^{\bullet , \bullet }))$ computes $H^ n(RF(K^\bullet ))$ we find the final assertion of the lemma holds true.
Computation of the first spectral sequence. We have ${}'E_1^{p, q} = H^ q(L^{p, \bullet })$ in other words
\[ {}'E_1^{p, q} = H^ q(F(I^{p, \bullet })) = R^ qF(K^ p) \]
as desired. Observe for later use that the maps ${}'d_1^{p, q} : {}'E_1^{p, q} \to {}'E_1^{p + 1, q}$ are the maps $R^ qF(K^ p) \to R^ qF(K^{p + 1})$ induced by $K^ p \to K^{p + 1}$ and the fact that $R^ qF$ is a functor.
Computation of the second spectral sequence. We have ${}''E_1^{p, q} = H^ q(L^{\bullet , p}) = H^ q(F(I^{\bullet , p}))$. Note that the complex $I^{\bullet , p}$ is bounded below, consists of injectives, and moreover each kernel, image, and cohomology group of the differentials is an injective object of $\mathcal{A}$. Hence we can split the differentials, i.e., each differential is a split surjection onto a direct summand. It follows that the same is true after applying $F$. Hence ${}''E_1^{p, q} = F(H^ q(I^{\bullet , p})) = F(H^ q_ I(I^{\bullet , p}))$. The differentials on this are $(-1)^ q$ times $F$ applied to the differential of the complex $H^ p_ I(I^{\bullet , \bullet })$ which is an injective resolution of $H^ p(K^\bullet )$. Hence the description of the $E_2$ terms.
$\square$
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