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Loyalty of the affine line in motivic homotopy theory
Abstract
We show that integers act invertibly on a derived scheme S if and only if they act invertibly on the reduced motivic suspension spectrum of the affine line in MSS. As a consequence, over regular locally noetherian schemes and derived schemes on which some nonzero integer acts as zero, the motivic sphere spectrum is A1-invariant after inverting primes that are not invertible in the base scheme. In particular, over regular locally noetherian Q-schemes, the motivic sphere spectrum is A1-invariant.
Related
- On the internal homotopy theory of motivic categories
- Schematic Functorialities of Birational Motivic Homotopy Categories
- The zeroth stable homotopy groups of motivic spheres over the integers
- Compact quotients of homogeneous spaces and homotopy theory of sphere bundles
- Metastable homotopy theory via Tate coalgebras
BibTeX
@misc{burghardt2026loyalty,
title = {{Loyalty of the affine line in motivic homotopy theory}},
author = {Viktor Burghardt},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11802},
eprint = {2610.11802},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11802},
}
RIS
TY - UNPB
TI - Loyalty of the affine line in motivic homotopy theory
AU - Viktor Burghardt
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11802
UR - https://arxiv.org/abs/2610.11802
AB - We show that integers act invertibly on a derived scheme S if and only if they act invertibly on the reduced motivic suspension spectrum of the affine line in MS_S. As a consequence, over regular locally noetherian schemes and derived schemes on which some nonzero integer acts as zero, the motivic sphere spectrum is A^1-invariant after inverting primes that are not invertible in the base scheme. In particular, over regular locally noetherian Q-schemes, the motivic sphere spectrum is A^1-invariant.
ER -