Lemma 15.63.16. Let $R$ be a ring. Let $f_1, \ldots , f_ r \in R$ be elements which generate the unit ideal. Let $a, b \in \mathbf{Z}$. Let $K^\bullet $ be a complex of $R$-modules. If for each $i$ the complex $K^\bullet \otimes _ R R_{f_ i}$ has tor amplitude in $[a, b]$, then $K^\bullet $ has tor amplitude in $[a, b]$.

**Proof.**
This follows immediately from Lemma 15.63.15 but can also be seen directly as follows. Note that $- \otimes _ R R_{f_ i}$ is an exact functor and that therefore

and similarly for every $R$-module $M$ we have

Hence the result follows from the fact that an $R$-module $N$ is zero if and only if $N_{f_ i}$ is zero for each $i$, see Algebra, Lemma 10.22.2. $\square$

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