$\mathbb A^1$-connected components of affine quadrics

TL;DR

This paper proves that for any smooth affine quadric $Q^{\psi}$ in $\mathbb{A}^n_k$, the field-valued $A^1$-connected components stabilize after two iterations of the naive functor $\mathcal{S}$, i.e., $\pi_0^{A^1}(Q^{\psi})(F)=\mathcal{S}^2(Q^{\psi})(F)$ for all finitely generated field extensions $F/k$. It provides explicit descriptions in the isotropic case via $\mathcal{S}(Q^{\psi})(F)\simeq F^*/\langle D(\phi_F)\rangle$ and shows stabilization to $\mathcal{S}^2(Q^{\psi})(F)$ in the anisotropic case, with $2$-ghost homotopy criteria governing the equivalence of points. The main result yields a complete $A^1$-connectedness criterion for affine quadrics in terms of Witt indices: $i_0(\varphi)\ge 2$ or $i_0(\varphi)=1$, $i_0(\psi)=0$, and $i_1(\psi)\ge 2$, where $\varphi=\psi\perp\langle-1\rangle$. This connects $A^1$-connectedness to classical quadratic-form invariants and extends the understanding of $A^1$-invariants for quadrics beyond the projective case.

Abstract

For any smooth quadratic hypersurface $X$ in $\mathbb A^n_k$, we use the iterations of the functor of naive $\mathbb{A}^1$-connected components $\mathcal{S}$ to study the field-valued sections of the sheaf of $\mathbb{A}^1$-connected components $π_0^{\mathbb{A}^1}(X)$ of $X$. We prove that for any field $F/k$, the canonical isomorphism $π_0^{\mathbb{A}^1}(X)(F) \xrightarrow{\sim} \lim_{n} \mathcal{S}^n(X)(F)$ stabilizes at $n=2$, meaning that $π_0^{\mathbb{A}^1}(X)(F)=\mathcal{S}^2(X)(F)$. Furthermore, by combining this result with Morel's characterization of $\mathbb{A}^1$-connected spaces in terms of the triviality of field-valued sections of $π_0^{\mathbb{A}^1}$, we provide a complete characterization of $\mathbb{A}^1$-connected smooth quadratic hypersurfaces in $\mathbb{A}^n_k$.

Theorems & Definitions (66)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Lemma 2.9
• Lemma 2.10