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.

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