The Stacks project

Lemma 35.8.9. Let $S$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $S$. Let $\tau $, $\mathcal{C}$, $U$, $\mathcal{U}$ be as in Lemma 35.8.8. Then there is an isomorphism of complexes

\[ \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}^ a) \cong s((A/R)_\bullet \otimes _ R M) \]

(see Section 35.3) where $R = \Gamma (U, \mathcal{O}_ U)$, $M = \Gamma (U, \mathcal{F}^ a)$ and $R \to A$ is a faithfully flat ring map. In particular

\[ \check{H}^ p(\mathcal{U}, \mathcal{F}^ a) = 0 \]

for all $p \geq 1$.

Proof. By Lemma 35.8.8 we see that $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}^ a)$ is isomorphic to $\check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{F}^ a)$ where $\mathcal{V} = \{ V \to U\} $ with $V = \coprod _{i = 1, \ldots n} U_ i$ affine also. Set $A = \Gamma (V, \mathcal{O}_ V)$. Since $\{ V \to U\} $ is a $\tau $-covering we see that $R \to A$ is faithfully flat. On the other hand, by definition of $\mathcal{F}^ a$ we have that the degree $p$ term $\check{\mathcal{C}}^ p(\mathcal{V}, \mathcal{F}^ a)$ is

\[ \Gamma (V \times _ U \ldots \times _ U V, \mathcal{F}^ a) = \Gamma (\mathop{\mathrm{Spec}}(A \otimes _ R \ldots \otimes _ R A), \mathcal{F}^ a) = A \otimes _ R \ldots \otimes _ R A \otimes _ R M \]

We omit the verification that the maps of the Čech complex agree with the maps in the complex $s((A/R)_\bullet \otimes _ R M)$. The vanishing of cohomology is Lemma 35.3.6. $\square$


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