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I am trying to understand if there is a good notion of a $K$-theory attached to the etale topology on a nice scheme $X$ (say smooth projective goem connected curve over a finite field is enough for me)?

I am aware of the etale algebraic K-theory, which is an etale sheafification of the algebraic K-theory presheaf, and I think it is not the same as $K$-specturm of the exact category of $l$-adic sheaves on X.

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  • $\begingroup$ What is the category of l-adic sheaves? If you are not careful, there might be an Eilenberg swindle. $\endgroup$
    – Z. M
    Commented Dec 2, 2022 at 20:26

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