The Stacks project

Lemma 47.13.4. With notation as above, assume $A \to B$ is a finite ring map of Noetherian rings. Then $R\mathop{\mathrm{Hom}}\nolimits (B, -)$ maps $D^+_{\textit{Coh}}(A)$ into $D^+_{\textit{Coh}}(B)$.

Proof. We have to show: if $K \in D^+(A)$ has finite cohomology modules, then the complex $R\mathop{\mathrm{Hom}}\nolimits (B, K)$ has finite cohomology modules too. This follows for example from Lemma 47.13.3 if we can show the ext modules $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(B, K)$ are finite $A$-modules. Since $K$ is bounded below there is a convergent spectral sequence

\[ \mathop{\mathrm{Ext}}\nolimits ^ p_ A(B, H^ q(K)) \Rightarrow \text{Ext}^{p + q}_ A(B, K) \]

This finishes the proof as the modules $\mathop{\mathrm{Ext}}\nolimits ^ p_ A(B, H^ q(K))$ are finite by Algebra, Lemma 10.71.9. $\square$

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Your email address will not be published.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 0A72. Beware of the difference between the letter 'O' and the digit '0'.