Vol. 7, No. 2, 2022

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Remarks on iterations of the $\mathbb A^1$-chain connected components construction

Chetan Balwe, Bandna Rani and Anand Sawant

Vol. 7 (2022), No. 2, 385–394
Abstract

We show that the sheaf of 𝔸1-connected components of a Nisnevich sheaf of sets and its universal 𝔸1-invariant quotient (obtained by iterating the 𝔸1-chain connected components construction and taking the direct limit) agree on field-valued points. This establishes an explicit formula for the field-valued points of the sheaf of 𝔸1-connected components of any space. Given any natural number  n, we construct an 𝔸1-connected space on which the iterations of the naive 𝔸1-connected components construction do not stabilize before the n-th stage.

Keywords
$\mathbb A^1$-connected components, $\mathbb A^1$-chain connected components, Morel's conjecture
Mathematical Subject Classification
Primary: 14F42
Milestones
Received: 21 July 2021
Revised: 1 February 2022
Accepted: 17 February 2022
Published: 13 September 2022
Authors
Chetan Balwe
Department of Mathematical Sciences
Indian Institute of Science Education and Research Mohali
Knowledge City, Sector 81
Mohali
India
Bandna Rani
Department of Mathematical Sciences
Indian Institute of Science Education and Research Mohali
Knowledge City, Sector 81
Mohali
India
Anand Sawant
School of Mathematics
Tata Institute of Fundamental Research
Mumbai
India