The Stacks project

Lemma 21.44.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{E}^\bullet $ be a bounded above complex of flat $\mathcal{O}$-modules with tor-amplitude in $[a, b]$. Then $\mathop{\mathrm{Coker}}(d_{\mathcal{E}^\bullet }^{a - 1})$ is a flat $\mathcal{O}$-module.

Proof. As $\mathcal{E}^\bullet $ is a bounded above complex of flat modules we see that $\mathcal{E}^\bullet \otimes _\mathcal {O} \mathcal{F} = \mathcal{E}^\bullet \otimes _\mathcal {O}^{\mathbf{L}} \mathcal{F}$ for any $\mathcal{O}$-module $\mathcal{F}$. Hence for every $\mathcal{O}$-module $\mathcal{F}$ the sequence

\[ \mathcal{E}^{a - 2} \otimes _\mathcal {O} \mathcal{F} \to \mathcal{E}^{a - 1} \otimes _\mathcal {O} \mathcal{F} \to \mathcal{E}^ a \otimes _\mathcal {O} \mathcal{F} \]

is exact in the middle. Since $\mathcal{E}^{a - 2} \to \mathcal{E}^{a - 1} \to \mathcal{E}^ a \to \mathop{\mathrm{Coker}}(d^{a - 1}) \to 0$ is a flat resolution this implies that $\text{Tor}_1^\mathcal {O}(\mathop{\mathrm{Coker}}(d^{a - 1}), \mathcal{F}) = 0$ for all $\mathcal{O}$-modules $\mathcal{F}$. This means that $\mathop{\mathrm{Coker}}(d^{a - 1})$ is flat, see Lemma 21.18.14. $\square$


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