The Stacks project

Lemma 20.38.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{U} : X = \bigcup _{i \in I} U_ i$ be a finite open covering. Let $\mathcal{F}^\bullet $ be a complex of $\mathcal{O}_ X$-modules. Let $\mathcal{B}$ be a set of open subsets of $X$. Assume

  1. every open in $X$ has a covering whose members are elements of $\mathcal{B}$,

  2. we have $U_{i_0\ldots i_ p} \in \mathcal{B}$ for all $i_0, \ldots , i_ p \in I$,

  3. for every $U \in \mathcal{B}$ and $p > 0$ we have

    1. $H^ p(U, \mathcal{F}^ q) = 0$,

    2. $H^ p(U, \mathop{\mathrm{Coker}}(\mathcal{F}^{q - 1} \to \mathcal{F}^ q)) = 0$, and

    3. $H^ p(U, H^ q(\mathcal{F})) = 0$.

Then the map

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

of Lemma 20.38.1 is an isomorphism in $D(\textit{Ab})$.

Proof. First assume $\mathcal{F}^\bullet $ is bounded below. In this case the map

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

is a quasi-isomorphism by Lemma 20.23.6. Namely, the map of double complexes $\check{\mathcal{C}}^\bullet _{alt}(\mathcal{U}, \mathcal{F}^\bullet ) \to \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}^\bullet )$ induces an isomorphism between the first pages of the second spectral sequences associated to these complexes (by Homology, Lemma 12.25.1) and these spectral sequences converge (Homology, Lemma 12.25.3). Thus the conclusion in this case by Lemma 20.25.2 and assumption (3)(a).

In general, by assumption (3)(c) we may choose a resolution $\mathcal{F}^\bullet \to \mathcal{I}^\bullet = \mathop{\mathrm{lim}}\nolimits \mathcal{I}_ n^\bullet $ as in Lemma 20.37.1. Then the map of the lemma becomes

\[ \mathop{\mathrm{lim}}\nolimits _ n \text{Tot}(\check{\mathcal{C}}^\bullet _{alt}(\mathcal{U}, \tau _{\geq -n}\mathcal{F}^\bullet )) \longrightarrow \Gamma (X, \mathcal{I}^\bullet ) = \mathop{\mathrm{lim}}\nolimits _ n \Gamma (X, \mathcal{I}_ n^\bullet ) \]

Here the arrow is in the derived category, but the equality on the right holds on the level of complexes. Note that (3)(b) shows that $\tau _{\geq -n}\mathcal{F}^\bullet $ is a bounded below complex satisfying the hypothesis of the lemma. Thus the case of bounded below complexes shows each of the maps

\[ \text{Tot}(\check{\mathcal{C}}^\bullet _{alt}(\mathcal{U}, \tau _{\geq -n}\mathcal{F}^\bullet )) \longrightarrow \Gamma (X, \mathcal{I}_ n^\bullet ) \]

is a quasi-isomorphism. The cohomologies of the complexes on the left hand side in given degree are eventually constant (as the alternating Čech complex is finite). Hence the same is true on the right hand side. Thus the cohomology of the limit on the right hand side is this constant value by Homology, Lemma 12.31.7 (or the stronger More on Algebra, Lemma 15.86.2) and we win. $\square$

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