Lemma 15.66.20. Let $R' \to R$ be a surjective ring map whose kernel is a nilpotent ideal. Let $K' \in D(R')$ and set $K = K' \otimes _{R'}^\mathbf {L} R$. Let $a, b \in \mathbf{Z}$. Then $K$ has tor amplitude in $[a, b]$ if and only if $K'$ does.

Proof. One direction follows from Lemma 15.66.13. For the other, assume $K$ has tor amplitude in $[a, b]$ and let $M'$ be an $R'$-module. We have to show that $K' \otimes _{R'}^\mathbf {L} M'$ has nonzero cohomology only for degrees contained in the interval $[a, b]$.

Let $I = \mathop{\mathrm{Ker}}(R' \to R)$. Then $I^ n = 0$ for some $n$. If $IM' = 0$, then we can view $M'$ as an $R$-module and argue as follows

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

which has nonvanishing cohomology only in the interval $[a, b]$ by assumption on $K$. If $I^{t + 1}M' = 0$, then we consider the short exact sequence

$0 \to IM' \to M' \to M'/IM' \to 0$

By induction on $t$ we have that both $K' \otimes _{R'}^\mathbf {L} IM'$ and $K' \otimes _{R'}^\mathbf {L} M'/IM'$ have nonzero cohomology only for degrees in the interval $[a, b]$. Then the distinguished triangle

$K' \otimes _{R'}^\mathbf {L} IM' \to K' \otimes _{R'}^\mathbf {L} M' K' \otimes _{R'}^\mathbf {L} M'/IM' (K' \otimes _{R'}^\mathbf {L} IM')[1]$

proves the same is true for $K' \otimes _{R'}^\mathbf {L} M'$ as desired. $\square$

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