The Stacks project

Remark 36.39.2. The construction of Lemma 36.39.1 is compatible with pullbacks. More precisely, given a morphism $f : X \to Y$ of schemes and a perfect object $K$ of $D(\mathcal{O}_ Y)$ of tor-amplitude in $[-1, 0]$ then $Lf^*K$ is a perfect object $K$ of $D(\mathcal{O}_ X)$ of tor-amplitude in $[-1, 0]$ and we have a canonical identification

\[ f^*\det (K) \longrightarrow \det (Lf^*K) \]

Moreover, if $K$ has rank $0$, then $\delta (K)$ pulls back to $\delta (Lf^*K)$ via this map. This is clear from the affine local construction of the determinant.

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Your email address will not be published.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



View Remark 36.39.2 as pdf