Lemma 15.65.17. Let $R$ be a ring. Let $a, b \in \mathbf{Z}$. Let $K^\bullet$ be a complex of $R$-modules. Let $R \to R'$ be a faithfully flat ring map. If the complex $K^\bullet \otimes _ R R'$ has tor amplitude in $[a, b]$, then $K^\bullet$ has tor amplitude in $[a, b]$.

Proof. Let $M$ be an $R$-module. Since $R \to R'$ is flat we see that

$(M \otimes _ R^{\mathbf{L}} K^\bullet ) \otimes _ R R' = ((M \otimes _ R R') \otimes _{R'}^{\mathbf{L}} (K^\bullet \otimes _ R R')$

and taking cohomology commutes with tensoring with $R'$. Hence $\text{Tor}_ i^ R(M, K^\bullet ) = \text{Tor}_ i^{R'}(M \otimes _ R R', K^\bullet \otimes _ R R')$. Since $R \to R'$ is faithfully flat, the vanishing of $\text{Tor}_ i^{R'}(M \otimes _ R R', K^\bullet \otimes _ R R')$ for $i \not\in [a, b]$ implies the same thing for $\text{Tor}_ i^ R(M, K^\bullet )$. $\square$

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