Lemma 15.77.4. Let $R$ be a ring. Let $K$ be a pseudo-coherent object of $D(R)$. Let $a, b \in \mathbf{Z}$. The following are equivalent

$K$ has projective-amplitude in $[a, b]$,

$K$ is perfect of tor-amplitude in $[a, b]$,

$\mathop{\mathrm{Ext}}\nolimits ^ i_ R(K, N) = 0$ for all finitely presented $R$-modules $N$ and all $i \not\in [-b, -a]$,

$H^ n(K) = 0$ for $n > b$ and $\mathop{\mathrm{Ext}}\nolimits ^ i_ R(K, N) = 0$ for all finitely presented $R$-modules $N$ and all $i > -a$, and

$H^ n(K) = 0$ for $n \not\in [a - 1, b]$ and $\mathop{\mathrm{Ext}}\nolimits ^{-a + 1}_ R(K, N) = 0$ for all finitely presented $R$-modules $N$.

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