\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*}

The Stacks project

Lemma 20.26.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{U} : X = \bigcup _{i \in I} U_ i$ be an open covering. For a bounded below complex $\mathcal{F}^\bullet $ of $\mathcal{O}_ X$-modules there is a canonical map

\[ \text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}^\bullet )) \longrightarrow R\Gamma (X, \mathcal{F}^\bullet ) \]

functorial in $\mathcal{F}^\bullet $ and compatible with (20.26.0.1) and (20.26.0.2). There is a spectral sequence $(E_ r, d_ r)_{r \geq 0}$ with

\[ E_2^{p, q} = H^ p(\text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \underline{H}^ q(\mathcal{F}^\bullet ))) \]

converging to $H^{p + q}(X, \mathcal{F}^\bullet )$.

Proof. Let ${\mathcal I}^\bullet $ be a bounded below complex of injectives. The map (20.26.0.1) for $\mathcal{I}^\bullet $ is a map $\Gamma (X, {\mathcal I}^\bullet ) \to \text{Tot}(\check{\mathcal{C}}^\bullet ({\mathcal U}, {\mathcal I}^\bullet ))$. This is a quasi-isomorphism of complexes of abelian groups as follows from Homology, Lemma 12.22.7 applied to the double complex $\check{\mathcal{C}}^\bullet ({\mathcal U}, {\mathcal I}^\bullet )$ using Lemma 20.12.1. Suppose ${\mathcal F}^\bullet \to {\mathcal I}^\bullet $ is a quasi-isomorphism of ${\mathcal F}^\bullet $ into a bounded below complex of injectives. Since $R\Gamma (X, {\mathcal F}^\bullet )$ is represented by the complex $\Gamma (X, {\mathcal I}^\bullet )$ we obtain the map of the lemma using

\[ \text{Tot}(\check{\mathcal{C}}^\bullet ({\mathcal U}, {\mathcal F}^\bullet )) \longrightarrow \text{Tot}(\check{\mathcal{C}}^\bullet ({\mathcal U}, {\mathcal I}^\bullet )). \]

We omit the verification of functoriality and compatibilities. To construct the spectral sequence of the lemma, choose a Cartan-Eilenberg resolution $\mathcal{F}^\bullet \to \mathcal{I}^{\bullet , \bullet }$, see Derived Categories, Lemma 13.21.2. In this case $\mathcal{F}^\bullet \to \text{Tot}(\mathcal{I}^{\bullet , \bullet })$ is an injective resolution and hence

\[ \text{Tot}(\check{\mathcal{C}}^\bullet ({\mathcal U}, \text{Tot}({\mathcal I}^{\bullet , \bullet }))) \]

computes $R\Gamma (X, \mathcal{F}^\bullet )$ as we've seen above. By Homology, Remark 12.22.8 we can view this as the total complex associated to the triple complex $\check{\mathcal{C}}^\bullet ({\mathcal U}, {\mathcal I}^{\bullet , \bullet })$ hence, using the same remark we can view it as the total complex associate to the double complex $A^{\bullet , \bullet }$ with terms

\[ A^{n, m} = \bigoplus \nolimits _{p + q = n} \check{\mathcal{C}}^ p({\mathcal U}, \mathcal{I}^{q, m}) \]

Since $\mathcal{I}^{q, \bullet }$ is an injective resolution of $\mathcal{F}^ q$ we can apply the first spectral sequence associated to $A^{\bullet , \bullet }$ (Homology, Lemma 12.22.4) to get a spectral sequence with

\[ E_1^{n, m} = \bigoplus \nolimits _{p + q = n} \check{\mathcal{C}}^ p(\mathcal{U}, \underline{H}^ m(\mathcal{F}^ q)) \]

which is the $n$th term of the complex $\text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \underline{H}^ m(\mathcal{F}^\bullet ))$. Hence we obtain $E_2$ terms as described in the lemma. Convergence by Homology, Lemma 12.22.6. $\square$


Comments (2)

Comment #3217 by Ingo Blechschmidt on

It appears that an analog of Tag 08C2 holds for the situation of this tag, especially because the proof of Tag 08C2 refers to an (unwritten) consequence of this one. Do you want me to prepare a pull request adding this analog?

Comment #3319 by on

Sure, go ahead. Sorry for the late reaction. In general this kind of thing is very helpful. Thanks.

There are also:

  • 2 comment(s) on Section 20.26: Čech cohomology of complexes

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