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arXiv · 1312.6388

A^1-connected components of schemes

Chetan Balwe·Amit Hogadi·Anand Sawant
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Abstract

A conjecture of Morel asserts that the sheaf of A^1-connected components of a simplicial sheaf X is A^1-invariant. A conjecture of Asok-Morel asserts that A^1-connected components of smooth k-schemes coincide with their A^1-chain-connected components and are birational invariants of smooth proper schemes. In this article, we exhibit examples of schemes for which Asok-Morel's conjectures fail to hold and whose Sing_* is not A^1-local. We also give equivalent conditions for Morel's conjecture to hold. A method suggested by these results is then used to prove Morel's conjecture for non-uniruled surfaces over a field k.

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Chetan Balwe, Amit Hogadi, Anand Sawant. 2013-12-22. A^1-connected components of schemes. https://arxiv.org/abs/1312.6388

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