Homology, Homotopy and Applications

Volume 23 (2021)

Number 1

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

Pages: 257 – 274

DOI: https://dx.doi.org/10.4310/HHA.2021.v23.n1.a14


Eoin Mackall (Mathematics and Statistics, University of Victoria, British Columbia, Canada)


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.


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

2010 Mathematics Subject Classification

14C25, 14C35

Copyright © 2020, Eoin Mackall. Permission to copy for private use granted.

Received 4 April 2020

Received revised 2 July 2020

Accepted 7 July 2020

Published 14 October 2020