Thomason’s Theorem for Varieties over Algebraically Closed Fields

We present a novel proof of Thomason’s theorem relating Bott inverted algebraic K-theory with finite coefficients and étale cohomology for smooth varieties over algebraically closed ground fields. Our proof involves first introducing a new theory, which we term algebraic K-homology, and proving it satisfies étale descent (with finite coefficients) on the category of normal, Cohen-Macaulay varieties. Then, we prove algebraic K-homology and algebraic K-theory (each taken with finite coefficients) coincide on smooth varieties upon inverting the Bott element.