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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