The Stacks project

Lemma 15.77.2. Let $R$ be a ring. Let $a, b \in \mathbf{Z}$. Let $K^\bullet $ be a pseudo-coherent complex of $R$-modules. The following are equivalent

  1. $K^\bullet $ is perfect with tor amplitude in $[a, b]$,

  2. for every prime $\mathfrak p$ we have $H^ i(K^\bullet \otimes _ R^{\mathbf{L}} \kappa (\mathfrak p)) = 0$ for all $i \not\in [a, b]$, and

  3. for every maximal ideal $\mathfrak m$ we have $H^ i(K^\bullet \otimes _ R^{\mathbf{L}} \kappa (\mathfrak m)) = 0$ for all $i \not\in [a, b]$.

Proof. We omit the proof of the implications (1) $\Rightarrow $ (2) $\Rightarrow $ (3). Assume (3). Let $i \in \mathbf{Z}$ with $i \not\in [a, b]$. By Lemma 15.76.4 we see that the assumption implies that $H^ i(K^\bullet )_{\mathfrak m} = 0$ for all maximal ideals of $R$. Hence $H^ i(K^\bullet ) = 0$, see Algebra, Lemma 10.23.1. Moreover, Lemma 15.76.4 now also implies that for every maximal ideal $\mathfrak m$ there exists an element $f \in R$, $f \not\in \mathfrak m$ such that $K^\bullet \otimes _ R R_ f$ is perfect with tor amplitude in $[a, b]$. Hence we conclude by appealing to Lemmas 15.74.12 and 15.66.16. $\square$

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