Lemma 21.46.4 (0F1M)—The Stacks project

Lemma 21.46.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$ . Let $a \in \mathbf{Z}$ . The following are equivalent

$E$ has tor-amplitude in $[a, \infty ]$ .
$E$ can be represented by a K-flat complex $\mathcal{E}^\bullet$ of flat $\mathcal{O}$ -modules with $\mathcal{E}^ i = 0$ for $i \not\in [a, \infty ]$ .

Moreover, we can choose $\mathcal{E}^\bullet$ such that any pullback by a morphism of ringed sites is a K-flat complex with flat terms.

Proof. The implication (2) $\Rightarrow$ (1) is immediate. Assume (1) holds. First we choose a K-flat complex $\mathcal{K}^\bullet$ with flat terms representing $E$ , see Lemma 21.17.11. For any $\mathcal{O}$ -module $\mathcal{M}$ the cohomology of

$\mathcal{K}^{n - 1} \otimes _\mathcal {O} \mathcal{M} \to \mathcal{K}^ n \otimes _\mathcal {O} \mathcal{M} \to \mathcal{K}^{n + 1} \otimes _\mathcal {O} \mathcal{M}$

computes $H^ n(E \otimes _\mathcal {O}^\mathbf {L} \mathcal{M})$ . This is always zero for $n < a$ . Hence if we apply Lemma 21.46.2 to the complex $\ldots \to \mathcal{K}^{a - 1} \to \mathcal{K}^ a \to \mathcal{K}^{a + 1}$ we conclude that $\mathcal{N} = \mathop{\mathrm{Coker}}(\mathcal{K}^{a - 1} \to \mathcal{K}^ a)$ is a flat $\mathcal{O}$ -module. We set

$\mathcal{E}^\bullet = \tau _{\geq a}\mathcal{K}^\bullet = (\ldots \to 0 \to \mathcal{N} \to \mathcal{K}^{a + 1} \to \ldots )$

The kernel $\mathcal{L}^\bullet$ of $\mathcal{K}^\bullet \to \mathcal{E}^\bullet$ is the complex

$\mathcal{L}^\bullet = (\ldots \to \mathcal{K}^{a - 1} \to \mathcal{I} \to 0 \to \ldots )$

where $\mathcal{I} \subset \mathcal{K}^ a$ is the image of $\mathcal{K}^{a - 1} \to \mathcal{K}^ a$ . Since we have the short exact sequence $0 \to \mathcal{I} \to \mathcal{K}^ a \to \mathcal{N} \to 0$ we see that $\mathcal{I}$ is a flat $\mathcal{O}$ -module. Thus $\mathcal{L}^\bullet$ is a bounded above complex of flat modules, hence K-flat by Lemma 21.17.8. It follows that $\mathcal{E}^\bullet$ is K-flat by Lemma 21.17.7.

Proof of the final assertion. Let $f : (\mathcal{C}', \mathcal{O}') \to (\mathcal{C}, \mathcal{O})$ be a morphism of ringed sites. By Lemma 21.18.1 the complex $f^*\mathcal{K}^\bullet$ is K-flat with flat terms. The complex $f^*\mathcal{L}^\bullet$ is K-flat as it is a bounded above complex of flat $\mathcal{O}'$ -modules. We have a short exact sequence of complexes of $\mathcal{O}'$ -modules

$0 \to f^*\mathcal{L}^\bullet \to f^*\mathcal{K}^\bullet \to f^*\mathcal{E}^\bullet \to 0$

because the short exact sequence $0 \to \mathcal{I} \to \mathcal{K}^ a \to \mathcal{N} \to 0$ of flat modules pulls back to a short exact sequence. By Lemma 21.17.7. the complex $f^*\mathcal{E}^\bullet$ is K-flat and the proof is complete. $\square$

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