The Stacks project

Lemma 15.77.6. Let $A \to B$ be a local ring homomorphism. Let $a, b \in \mathbf{Z}$. Let $d \geq 0$. Let $K^\bullet $ be a complex of $B$-modules. Assume

  1. the ring map $A \to B$ is flat,

  2. the ring $B/\mathfrak m_ AB$ is regular of dimension $d$,

  3. $K^\bullet $ is pseudo-coherent as a complex of $B$-modules, and

  4. $K^\bullet $ has tor amplitude in $[a, b]$ as a complex of $A$-modules, in fact it suffices if $H^ i(K^\bullet \otimes _ A^\mathbf {L} \kappa (\mathfrak m_ A))$ is nonzero only for $i \in [a, b]$.

Then $K^\bullet $ is perfect as a complex of $B$-modules with tor amplitude in $[a - d, b]$.

Proof. By (3) we may assume that $K^\bullet $ is a bounded above complex of finite free $B$-modules. We compute

\begin{align*} K^\bullet \otimes _ B^{\mathbf{L}} \kappa (\mathfrak m_ B) & = K^\bullet \otimes _ B \kappa (\mathfrak m_ B) \\ & = (K^\bullet \otimes _ A \kappa (\mathfrak m_ A)) \otimes _{B/\mathfrak m_ A B} \kappa (\mathfrak m_ B) \\ & = (K^\bullet \otimes _ A \kappa (\mathfrak m_ A)) \otimes ^{\mathbf{L}}_{B/\mathfrak m_ A B} \kappa (\mathfrak m_ B) \end{align*}

The first equality because $K^\bullet $ is a bounded above complex of flat $B$-modules. The second equality follows from basic properties of the tensor product. The third equality holds because $K^\bullet \otimes _ A \kappa (\mathfrak m_ A) = K^\bullet / \mathfrak m_ A K^\bullet $ is a bounded above complex of flat $B/\mathfrak m_ A B$-modules. Since $K^\bullet $ is a bounded above complex of flat $A$-modules by (1), the cohomology modules $H^ i$ of the complex $K^\bullet \otimes _ A \kappa (\mathfrak m_ A)$ are nonzero only for $i \in [a, b]$ by assumption (4). Thus the spectral sequence of Example 15.62.1 and the fact that $B/\mathfrak m_ AB$ has finite global dimension $d$ (by (2) and Algebra, Proposition 10.110.1) shows that $H^ j(K^\bullet \otimes _ B^{\mathbf{L}} \kappa (\mathfrak m_ B))$ is zero for $j \not\in [a - d, b]$. This finishes the proof by Lemma 15.77.2. $\square$


Comments (0)


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Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09PC. Beware of the difference between the letter 'O' and the digit '0'.