The Stacks project

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

\[ H^ i(K^\bullet )_{f_ i} = H^ i(K^\bullet ) \otimes _ R R_{f_ i} = H^ i(K^\bullet \otimes _ R R_{f_ i}). \]

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

\[ H^ i(K^\bullet \otimes _ R^{\mathbf{L}} M)_{f_ i} = H^ i(K^\bullet \otimes _ R^{\mathbf{L}} M) \otimes _ R R_{f_ i} = H^ i(K^\bullet \otimes _ R R_{f_ i} \otimes _{R_{f_ i}}^{\mathbf{L}} M_{f_ i}). \]

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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