Lemma 15.83.9. Let R be a ring. Let A = R[x_1, \ldots , x_ d]/I be flat and of finite presentation over R. Let \mathfrak q \subset A be a prime ideal lying over \mathfrak p \subset R. Let K \in D(A) be pseudo-coherent. Let a, b \in \mathbf{Z}. If H^ i(K_\mathfrak q \otimes _{R_\mathfrak p}^\mathbf {L} \kappa (\mathfrak p)) is nonzero only for i \in [a, b], then K_\mathfrak q has tor amplitude in [a - d, b] over R.
Proof. By Lemma 15.82.8 K is pseudo-coherent as a complex of R[x_1, \ldots , x_ d]-modules. Therefore we may assume A = R[x_1, \ldots , x_ d]. Applying Lemma 15.77.6 to R_\mathfrak p \to A_\mathfrak q and the complex K_\mathfrak q using our assumption, we find that K_\mathfrak q is perfect in D(A_\mathfrak q) with tor amplitude in [a - d, b]. Since R_\mathfrak p \to A_\mathfrak q is flat, we conclude by Lemma 15.66.11. \square
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