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arXiv · 2610.11802

Loyalty of the affine line in motivic homotopy theory

Viktor Burghardt
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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 $\mathrm{MS}_S$. As a consequence, over regular locally noetherian schemes and derived schemes on which some nonzero integer acts as zero, the motivic sphere spectrum is $\mathbf{A}^1$-invariant after inverting primes that are not invertible in the base scheme. In particular, over regular locally noetherian $\mathbf{Q}$-schemes, the motivic sphere spectrum is $\mathbf{A}^1$-invariant.

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Viktor Burghardt. 2026-10-08. Loyalty of the affine line in motivic homotopy theory. https://arxiv.org/abs/2610.11802

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