The Stacks project

Lemma 29.7.1. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Assume $X$ is quasi-compact and $X$ and $S$ have affine diagonal (e.g., if $X$ and $S$ are separated). In this case we can compute $Rf_*\mathcal{F}$ as follows:

  1. Choose a finite affine open covering $\mathcal{U} : X = \bigcup _{i = 1, \ldots , n} U_ i$.

  2. For $i_0, \ldots , i_ p \in \{ 1, \ldots , n\} $ denote $f_{i_0 \ldots i_ p} : U_{i_0 \ldots i_ p} \to S$ the restriction of $f$ to the intersection $U_{i_0 \ldots i_ p} = U_{i_0} \cap \ldots \cap U_{i_ p}$.

  3. Set $\mathcal{F}_{i_0 \ldots i_ p}$ equal to the restriction of $\mathcal{F}$ to $U_{i_0 \ldots i_ p}$.

  4. Set

    \[ \check{\mathcal{C}}^ p(\mathcal{U}, f, \mathcal{F}) = \bigoplus \nolimits _{i_0 \ldots i_ p} f_{i_0 \ldots i_ p *} \mathcal{F}_{i_0 \ldots i_ p} \]

    and define differentials $d : \check{\mathcal{C}}^ p(\mathcal{U}, f, \mathcal{F}) \to \check{\mathcal{C}}^{p + 1}(\mathcal{U}, f, \mathcal{F})$ as in Cohomology, Equation (20.9.0.1).

Then the complex $\check{\mathcal{C}}^\bullet (\mathcal{U}, f, \mathcal{F})$ is a complex of quasi-coherent sheaves on $S$ which comes equipped with an isomorphism

\[ \check{\mathcal{C}}^\bullet (\mathcal{U}, f, \mathcal{F}) \longrightarrow Rf_*\mathcal{F} \]

in $D^{+}(S)$. This isomorphism is functorial in the quasi-coherent sheaf $\mathcal{F}$.

Proof. Consider the resolution $\mathcal{F} \to {\mathfrak C}^\bullet (\mathcal{U}, \mathcal{F})$ of Cohomology, Lemma 20.24.1. We have an equality of complexes $\check{\mathcal{C}}^\bullet (\mathcal{U}, f, \mathcal{F}) = f_*{\mathfrak C}^\bullet (\mathcal{U}, \mathcal{F})$ of quasi-coherent $\mathcal{O}_ S$-modules. The morphisms $j_{i_0 \ldots i_ p} : U_{i_0 \ldots i_ p} \to X$ and the morphisms $f_{i_0 \ldots i_ p} : U_{i_0 \ldots i_ p} \to S$ are affine by Morphisms, Lemma 28.11.11 and Lemma 29.2.5. Hence $R^ qj_{i_0 \ldots i_ p *}\mathcal{F}_{i_0 \ldots i_ p}$ as well as $R^ qf_{i_0 \ldots i_ p *}\mathcal{F}_{i_0 \ldots i_ p}$ are zero for $q > 0$ (Lemma 29.2.3). Using $f \circ j_{i_0 \ldots i_ p} = f_{i_0 \ldots i_ p}$ and the spectral sequence of Cohomology, Lemma 20.13.8 we conclude that $R^ qf_*(j_{i_0 \ldots i_ p *}\mathcal{F}_{i_0 \ldots i_ p}) = 0$ for $q > 0$. Since the terms of the complex ${\mathfrak C}^\bullet (\mathcal{U}, \mathcal{F})$ are finite direct sums of the sheaves $j_{i_0 \ldots i_ p *}\mathcal{F}_{i_0 \ldots i_ p}$ we conclude using Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7) that

\[ Rf_* \mathcal{F} = f_*{\mathfrak C}^\bullet (\mathcal{U}, \mathcal{F}) = \check{\mathcal{C}}^\bullet (\mathcal{U}, f, \mathcal{F}) \]

as desired. $\square$


Comments (2)

Comment #938 by correction_bot on

  1. Why is there a reference to Lemma 20.14.8?

  2. "The terms of the complex are finite direct summands of the sheaves ", replace "summands" with "sums".

  3. Where is quasi-compactness of used?


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