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Affine representability results in $$\mathbb{A}^1$$-homotopy theory. I: Vector bundles. (English) Zbl 1401.14118
The main goal of this paper is to prove the representability of vector bundles in the (unstable) $$\mathbb{A}^1$$-homotopy category for smooth affine schemes over a base scheme for which the Bass-Quillen conjecture holds (for example if the base is the spectrum of a noetherian ring regular over a Dedekind domain with perfect residue fields, see Theorem 5.2.1). This result generalizes a theorem of Morel (Theorem 8.1 in [F. Morel, $$\mathbb A^1$$-algebraic topology over a field. Berlin: Springer (2012; Zbl 1263.14003)]) using a quite different method. The precise statement of the result is as follows: for any scheme $$X$$ and any integer $$r\geq0$$ denote by $$V_r(X)$$ the isomorphism classes of vector bundles over $$X$$. We say that the Bass-Quillen conjecture holds for a ring $$A$$ if for any $$n\geq0$$, the map $V_r(\text{Spec}\;A) \to V_r(\text{Spec}\;A[t_1,\cdots,t_n])$ is a bijection. Suppose that $$S$$ is a quasi-compact quasi-separated scheme such that for any scheme $$X$$ smooth affine over $$S$$, the Bass-Quillen conjecture holds for $$\mathcal{O}(X)$$. Then any scheme $$X$$ smooth affine over $$S$$, there is a canonical bijection $V_r(X)\simeq[X,Gr_r]_{\mathbf{H}(S)}$ where the right hand side is the homomorphisms in the $$\mathbb{A}^1$$-homotopy category and $$Gr_r$$ is the infinite Grassmannian. There are two crucial ingredients of the proof: the first is affine Nisnevich excision property introduced by Morel, which plays the role of the Mayer-Vietoris property for affine schemes; the second one is Theorem 4.2.3, which is originally due to M. Schlichting [Adv. Math. 320, 1–81 (2017; Zbl 1387.19002)]. The authors then deduce Corollary 4.2.4, which says that the $$\mathbb{A}^1$$ fibrant replacement functor $$\mathrm{Sing}^{\mathbb{A}^1}$$ preserves the affine Nisnevich excision property for simplicial presheaves whose $$\pi_0$$ presheaf is $$\mathbb{A}^1$$-invariant; the condition on $$\pi_0$$ being guaranteed by the Bass-Quillen conjecture, the authors conclude using Theorem 3.3.4, which compares affine Nisnevich excision and Nisnevich topology using Voevodsky’s theory of cd-structures.

##### MSC:
 14F42 Motivic cohomology; motivic homotopy theory 55R15 Classification of fiber spaces or bundles in algebraic topology
##### Keywords:
motivic homotopy theory; vector bundles
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