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.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)