Affine representability results in \(\mathbb{A}^1\)-homotopy theory. I: Vector bundles.

*(English)*Zbl 1401.14118The 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.

Reviewer: Fangzhou Jin (Essen)