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arXiv · 2103.02204

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

Eoin Mackall
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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.

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Eoin Mackall. 2021-03-03. $\mathbb{A}^1$-homotopy equivalences and a theorem of Whitehead. https://doi.org/10.4310/hha.2021.v23.n1.a14

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