The Stacks project

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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