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

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 ${\mathcal{S}}^n(X_n)\ne {\mathcal{S}}^{n+1}(X_n)$ and ${\mathcal{S}}^{n+2}(X_n)=\ast$.

15 pages, comments are welcome, v3: final version before page proofs, accepted for publication in NYJM