Lemma 15.66.15. Let $A \to B$ be a ring map. Let $K^\bullet $ be a complex of $B$-modules. Let $a, b \in \mathbf{Z}$. The following are equivalent

$K^\bullet $ has tor amplitude in $[a, b]$ as a complex of $A$-modules,

$K^\bullet _\mathfrak q$ has tor amplitude in $[a, b]$ as a complex of $A_\mathfrak p$-modules for every prime $\mathfrak q \subset B$ with $\mathfrak p = A \cap \mathfrak q$,

$K^\bullet _\mathfrak m$ has tor amplitude in $[a, b]$ as a complex of $A_\mathfrak p$-modules for every maximal ideal $\mathfrak m \subset B$ with $\mathfrak p = A \cap \mathfrak m$.

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