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.

中文翻译：

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$ \ mathbb {A} ^ 1 $-同等式和怀特海定理

我们证明了Chow组和Grothendieck组的相干绳轮的Whitehead定理的类似物（来自代数拓扑）：光滑射影变种之间的同态，其推论是Chow组或Grothendieck组的相干绳轮的同构，是同构。作为推论，我们证明了在光滑的投影变种之间没有非平凡的天真$ \ mathbb {A} ^ 1 $-同伦对等。