\( \mathbb{A}^1\)-homotopy equivalences and a theorem of Whitehead (Q2214752)

The author proves the following analogs of Whitehead's theorem. Given a smooth projective variety \(X\), let \(\mathrm{CH}(X):=\bigoplus_{i=0}^\infty\mathrm{CH}_i(X)\) be the Chow group of \(X\). Let \(G(X)\) be the Grothendieck group of coherent sheaves on \(X\). Then if \(f:X\to Y\) is a morphism of smooth projective varieties such that the proper pushforward \(f_*:\mathrm{CH}(X)\to\mathrm{CH}(Y)\) is an isomorphism, then \(f\) is an isomorphism. Similarly, if \(f_*:G(X)\to G(Y)\) is an isomorphism, then \(f\) is an isomorphism. These two theorems are proved using naïve \(\mathbb{A}^1\)-homotopy theory. A fun by-product of the article is Corollary 4.9, where the author proves that if two smooth projective varieties \(X,Y\) are naïvely \(\mathbb{A}^1\)-homotopic, then \(X\) and \(Y\) are isomorphic. The article is well-written and contains nice exposition, especially in Section 2. Naïve \(\mathbb{A}^1\)-homotopy theory has not been studied to the same extent as its non-naïve counterpart, so it is encouraging to see interesting results arising from this area. For another application of naïve \(\mathbb{A}^1\)-homotopy theory, see \textit{C. Cazanave}'s work on the global \(\mathbb{A}^1\)-degree of rational functions [C. R., Math., Acad. Sci. Paris 346, No. 3--4, 129--133 (2008; Zbl 1151.14016)].