The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

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:

  1. As $I^{p, \bullet }$ is an injective resolution of $K^ p$ we see that $RF$ is defined at $K^ p[0]$ with value $F(I^{p, \bullet })$.

  2. 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 $F(H^ p_ I(I^{\bullet , \bullet }))$.

  3. By Homology, Lemma 12.22.7 the total complex $sI^\bullet $ is an injective resolution of $K^\bullet $. Hence $RF$ is defined at $K^\bullet $ with value $F(sI^\bullet )$.

Consider the two spectral sequences associated to the double complex $L^{\bullet , \bullet } = F(I^{\bullet , \bullet })$, see Homology, Lemma 12.22.4. These are both bounded, converge to $H^*(sL^\bullet )$, and induce finite filtrations on $H^ n(sL^\bullet )$, see Homology, Lemma 12.22.6. Since $sL^\bullet = s(F(I^{\bullet , \bullet })) = F(sI^\bullet )$ 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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