The Stacks project

Remark 42.56.2. Let $X$ be a scheme such that $2$ is invertible on $X$. Then the Adams operator $\psi ^2$ can be defined on the $K$-group $K_0(X) = K_0(D_{perf}(\mathcal{O}_ X))$ (Derived Categories of Schemes, Definition 36.38.2) in a straightforward manner. Namely, given a perfect complex $L$ on $X$ we get an action of the group $\{ \pm 1\} $ on $L \otimes ^\mathbf {L} L$ by switching the factors. Then we can set

\[ \psi ^2(L) = [(L \otimes ^\mathbf {L} L)^+] - [(L \otimes ^\mathbf {L} L)^-] \]

where $(-)^+$ denotes taking invariants and $(-)^-$ denotes taking anti-invariants (suitably defined). Using exactness of taking invariants and anti-invariants one can argue similarly to the proof of Lemma 42.56.1 to show that this is well defined. When $2$ is not invertible on $X$ the situation is a good deal more complicated and another approach has to be used.

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Your email address will not be published.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



View Remark 42.56.2 as pdf