Iterations of the functor of naive $\mathbb A^1$-connected components of varieties

TL;DR

This work analyzes the stabilization of the universal $\mathbb A^1$-invariant quotient $\mathcal L(\mathcal F)$, expressed as $\mathcal L(\mathcal F)=\varinjlim_n \mathcal S^n(\mathcal F)$, for schemes over a field. It constructs, for each $n$, an $\mathbb A^1$-connected variety $X_n$ of dimension $n+1$ with $\mathcal S^n(X_n)\neq\mathcal S^{n+1}(X_n)$ and $\mathcal S^{n+2}(X_n)=*$, thereby proving that infinite iterations are essential even in the geometric category of varieties. The proof combines an explicit iterative construction of $X_n$ via $n$-fold fiber products of a base surface $X_1$ (defined by $y_1^2-x_0^2 f(x_1)=0$) with a careful analysis of $\mathbb A^1$-homotopies and Nisnevich gluing to produce controlled homotopies in the iterated naive components. It further establishes that $\pi_0^{\mathbb A^1}(X_n)=\mathcal S^{n+2}(X_n)=*$, placing the phenomenon within the broader non-stabilization literature and highlighting the limits of finite-stage approximations in $\mathbb A^1$-homotopy theory.

Abstract

For any sheaf of sets $\mathcal F$ on $Sm/k$, it is well known that the universal $\mathbb A^1$-invariant quotient of $\mathcal F$ is given as the colimit of sheaves $\mathcal S^n(\mathcal F)$ where $\mathcal S(F)$ is the sheaf of naive $\mathbb A^1$-connected components of $\mathcal F$. We show that these infinite iterations of naive $\mathbb A^1$-connected components in the construction of universal $\mathbb A^1$-invariant quotient for a scheme are certainly required. For every $n$, we construct an $\mathbb A^1$-connected variety $X_n$ such that $\mathcal S^n(X_n)\neq \mathcal S^{n+1}(X_n)$ and $\mathcal S^{n+2}(X_n)=*$.

Theorems & Definitions (35)

• Theorem 1.1
• Definition 2.2
• Definition 2.3
• Lemma 2.4
• proof
• Definition 2.5
• Definition 2.6
• Lemma 2.7
• Lemma 2.8
• Theorem 2.9