The Stacks project

Lemma 21.47.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$. Let $a \leq b$ be integers. If $E$ has tor amplitude in $[a, b]$ and is $(a - 1)$-pseudo-coherent, then $E$ is perfect.

Proof. Let $U$ be an object of $\mathcal{C}$. After replacing $U$ by the members of a covering and $\mathcal{C}$ by the localization $\mathcal{C}/U$ we may assume there exists a strictly perfect complex $\mathcal{E}^\bullet $ and a map $\alpha : \mathcal{E}^\bullet \to E$ such that $H^ i(\alpha )$ is an isomorphism for $i \geq a$. We may and do replace $\mathcal{E}^\bullet $ by $\sigma _{\geq a - 1}\mathcal{E}^\bullet $. Choose a distinguished triangle

\[ \mathcal{E}^\bullet \to E \to C \to \mathcal{E}^\bullet [1] \]

From the vanishing of cohomology sheaves of $E$ and $\mathcal{E}^\bullet $ and the assumption on $\alpha $ we obtain $C \cong \mathcal{K}[a - 2]$ with $\mathcal{K} = \mathop{\mathrm{Ker}}(\mathcal{E}^{a - 1} \to \mathcal{E}^ a)$. Let $\mathcal{F}$ be an $\mathcal{O}$-module. Applying $- \otimes _\mathcal {O}^\mathbf {L} \mathcal{F}$ the assumption that $E$ has tor amplitude in $[a, b]$ implies $\mathcal{K} \otimes _\mathcal {O} \mathcal{F} \to \mathcal{E}^{a - 1} \otimes _\mathcal {O} \mathcal{F}$ has image $\mathop{\mathrm{Ker}}(\mathcal{E}^{a - 1} \otimes _\mathcal {O} \mathcal{F} \to \mathcal{E}^ a \otimes _\mathcal {O} \mathcal{F})$. It follows that $\text{Tor}_1^\mathcal {O}(\mathcal{E}', \mathcal{F}) = 0$ where $\mathcal{E}' = \mathop{\mathrm{Coker}}(\mathcal{E}^{a - 1} \to \mathcal{E}^ a)$. Hence $\mathcal{E}'$ is flat (Lemma 21.17.15). Thus there exists a covering $\{ U_ i \to U\} $ such that $\mathcal{E}'|_{U_ i}$ is a direct summand of a finite free module by Modules on Sites, Lemma 18.29.3. Thus the complex

\[ \mathcal{E}'|_{U_ i} \to \mathcal{E}^{a - 1}|_{U_ i} \to \ldots \to \mathcal{E}^ b|_{U_ i} \]

is quasi-isomorphic to $E|_{U_ i}$ and $E$ is perfect. $\square$

Comments (1)

Comment #8698 by Shizhang on

Should be instead? Second-to-last line of the proof, the (a-1) in the upper-index should be (a+1) instead?

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