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 $S$.
