The Stacks project

Lemma 15.66.2. Let $R$ be a ring. Let $K^\bullet $ be a bounded above complex of flat $R$-modules with tor-amplitude in $[a, b]$. Then $\mathop{\mathrm{Coker}}(d_ K^{a - 1})$ is a flat $R$-module.

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

\[ K^{a - 2} \otimes _ R M \to K^{a - 1} \otimes _ R M \to K^ a \otimes _ R M \]

is exact in the middle. Since $K^{a - 2} \to K^{a - 1} \to K^ a \to \mathop{\mathrm{Coker}}(d_ K^{a - 1}) \to 0$ is a flat resolution this implies that $\text{Tor}_1^ R(\mathop{\mathrm{Coker}}(d_ K^{a - 1}), M) = 0$ for all $R$-modules $M$. This means that $\mathop{\mathrm{Coker}}(d_ K^{a - 1})$ is flat, see Algebra, Lemma 10.75.8. $\square$

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