Lemma 36.23.3 (0FMQ)—The Stacks project

Lemma 36.23.3. Let $f : X \to Y$ be a morphism of schemes with $Y = \mathop{\mathrm{Spec}}(A)$ affine. Let $\mathcal{U} : X = \bigcup _{i \in I} U_ i$ be a finite affine open covering such that all the finite intersections $U_{i_0 \ldots i_ p} = U_{i_0} \cap \ldots \cap U_{i_ p}$ are affine. Let $\mathcal{F}^\bullet$ be a bounded complex of $f^{-1}\mathcal{O}_ Y$ -modules. Assume for all $n \in \mathbf{Z}$ the sheaf $\mathcal{F}^ n$ is a flat $f^{-1}\mathcal{O}_ Y$ -module and $\mathcal{F}^ n$ has the structure of a quasi-coherent $\mathcal{O}_ X$ -module compatible with the given $p^{-1}\mathcal{O}_ Y$ -module structure (but the differentials in the complex $\mathcal{F}^\bullet$ need not be $\mathcal{O}_ X$ -linear). Then the complex $\text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}^\bullet ))$ is K-flat as a complex of $A$ -modules.

Proof. We may write

$\mathcal{F}^\bullet = (\mathcal{F}^ a \to \ldots \to \mathcal{F}^ b)$

Arguing by induction on $b - a$ and considering the distinguished triangle

$\mathcal{F}^ b[-b] \to (\mathcal{F}^ a \to \ldots \to \mathcal{F}^ b) \to (\mathcal{F}^ a \to \ldots \to \mathcal{F}^{b - 1}) \to \mathcal{F}^ b[-b + 1]$

and using More on Algebra, Lemma 15.59.5 we reduce to the case where $\mathcal{F}^\bullet$ consists of a single quasi-coherent $\mathcal{O}_ X$ -module $\mathcal{F}$ placed in degree $0$ . In this case the Čech complex for $\mathcal{F}$ and $\mathcal{U}$ is homotopy equivalent to the alternating Čech complex, see Cohomology, Lemma 20.23.6. Since $U_{i_0 \ldots i_ p}$ is always affine, we see that $\mathcal{F}(U_{i_0 \ldots i_ p})$ is $A$ -flat. Hence $\check{\mathcal{C}}_{alt}^\bullet (\mathcal{U}, \mathcal{F})$ is a bounded complex of flat $A$ -modules and hence K-flat by More on Algebra, Lemma 15.59.7. $\square$

The Stacks project

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