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It is known that cdh-sheafification of algebraic $K$-theory coincides with homotopy $K$-theory. Although I haven't gone through the details of the proof, I was wondering whether there is a general set of abstract conditions that let's you deduce $\mathbb{A}^1$-invariance from another set of seemingly unrelated properties (including cdh-descent)? (not just the algebraic $K$-theory case) For example let's say we have a motivic like cohomolgy but we can't prove the $\mathbb{A}^1$-invariance for it directly, is there a method to deduce it using another set of properties? (I'd also be interested in seeing some examples that this kind of technique has been applied outside of the specific algebraic $K$-theory setting)

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    $\begingroup$ If F is the sheaf representing your cohomology theory, then you have $A^1$-invariance for smooth schemes X if you have the Gysin triangle $F(X\times \infty) \to F(X \times P^1) \to F(X \times A^1)$ and the projective bundle formula. If F has cdh descent and you're in some fantasy world where every scheme is cdh-locally smooth (e.g. over a base field with resolution of singularities) then you get A^1-invariance on singular schemes as well. This is at least a heuristic explanation of why we might expect $K_{cdh}$ to satisfy A^1-invariance... $\endgroup$
    – crystalline
    Commented Jul 30, 2023 at 21:39

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