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

TL;DR

This paper establishes algebraic analogs of Whitehead's theorem for Chow groups and Grothendieck groups, showing that certain morphisms induce isomorphisms and that no nontrivial naive $A^1$-homotopy equivalences exist between smooth projective varieties.

Contribution

It proves algebraic versions of Whitehead's theorem for Chow and Grothendieck groups, and demonstrates the triviality of naive $A^1$-homotopy equivalences in this context.

Findings

Morphisms with isomorphic pushforwards induce isomorphisms on Chow and Grothendieck groups.

No nontrivial naive $A^1$-homotopy equivalences exist between smooth projective varieties.

Analogues of Whitehead's theorem are established in algebraic geometry.

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.