${\mathbb{A}}^1$-homotopy equivalences and a theorem of Whitehead

Abstract

We prove analogs of Whitehead's theorem (from algebraic topology) for both the Chow groups and for the Grothendieck group of coherent sheaves: a morphism between smooth projective varieties whose pushforward is an isomorphism on the Chow groups, or on the Grothendieck group of coherent sheaves, is an isomorphism. As a corollary, we show that there are no nontrivial naive ${\mathbb{A}}^1$-homotopy equivalences between smooth projective varieties.