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12 votes
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Rational homotopy invariance of algebraic $K$-theory

The theorem can be found in more general form in Land, Tamme On the K-theory of pullbacks, Lemma 2.4.
Lennart Meier's user avatar
5 votes
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Etale $K$ theory coincides with algebraic one in high enough degrees

To my knowledge the most general known statement has been proven by Clausen and Mathew in their paper Hyperdescent and étale K-theory as Theorem 1.2. The precise conditions on your commutative ring (o …
Lennart Meier's user avatar