The Stacks project

Lemma 21.44.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$. Let $a, b \in \mathbf{Z}$ with $a \leq b$. The following are equivalent

  1. $E$ has tor-amplitude in $[a, b]$.

  2. $E$ is represented by a complex $\mathcal{E}^\bullet $ of flat $\mathcal{O}$-modules with $\mathcal{E}^ i = 0$ for $i \not\in [a, b]$.

Proof. If (2) holds, then we may compute $E \otimes _\mathcal {O}^\mathbf {L} \mathcal{F} = \mathcal{E}^\bullet \otimes _\mathcal {O} \mathcal{F}$ and it is clear that (1) holds.

Assume that (1) holds. We may represent $E$ by a bounded above complex of flat $\mathcal{O}$-modules $\mathcal{K}^\bullet $, see Section 21.18. Let $n$ be the largest integer such that $\mathcal{K}^ n \not= 0$. If $n > b$, then $\mathcal{K}^{n - 1} \to \mathcal{K}^ n$ is surjective as $H^ n(\mathcal{K}^\bullet ) = 0$. As $\mathcal{K}^ n$ is flat we see that $\mathop{\mathrm{Ker}}(\mathcal{K}^{n - 1} \to \mathcal{K}^ n)$ is flat (Modules on Sites, Lemma 18.28.9). Hence we may replace $\mathcal{K}^\bullet $ by $\tau _{\leq n - 1}\mathcal{K}^\bullet $. Thus, by induction on $n$, we reduce to the case that $K^\bullet $ is a complex of flat $\mathcal{O}$-modules with $\mathcal{K}^ i = 0$ for $i > b$.

Set $\mathcal{E}^\bullet = \tau _{\geq a}\mathcal{K}^\bullet $. Everything is clear except that $\mathcal{E}^ a$ is flat which follows immediately from Lemma 21.44.2 and the definitions. $\square$


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