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

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.

Keywords

Chow group, Grothendieck group, $\mathbb{A}^1$-homotopy

2010 Mathematics Subject Classification

14C25, 14C35

Full Text (PDF format)

Received 4 April 2020

Received revised 2 July 2020

Accepted 7 July 2020

Published 14 October 2020