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

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

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

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

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