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Given a ringed topological space $S$ one can easily define its K and K'-theory as the K-theories of the categories of locally free sheaves and of the category of coherent sheaves on it, respectively. My question is: did anybody study the case when $S$ is not a scheme? Which conditions on $S$ ensure some sort of excision and Mayer-Vietoris for these theories?

I am mostly interested in the case when $S$ is a henselian or a formal scheme, and in $K$-theories with $\mathbb{Z}/l\mathbb{Z}$-coefficients, where $l$ is invertible on $S$.

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  • $\begingroup$ I'm wondering whether you finally found an answer for this question, especially for formal schemes? $\endgroup$
    – user127776
    Feb 9, 2021 at 16:06
  • $\begingroup$ I definitely don't remember any answer to this question at the moment.(: $\endgroup$ Feb 9, 2021 at 18:53

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