Voevodsky's motives and zeroth slice of motivic sphere
Abstract
Let S be a noetherian scheme of finite Krull dimension and let R be a commutative ring in which every positive residue characteristic of S is invertible. We prove that Voevodsky's stable Nisnevich category of motives with R-coefficients is canonically equivalent to modules over the zeroth slice of the motivic sphere with derived R-coefficients. The comparison identifies the corresponding commutative coefficient algebras and is compatible with arbitrary pullback. The theorem applies to singular and nonreduced schemes, including mixed-characteristic bases, and requires no excellence or finite-type hypothesis. Its special cases include rational coefficients over arbitrary bases, arbitrary coefficients over ℚ-schemes, and ℤ[1/p]-algebra coefficients over ℤ(p)-schemes. The proof establishes base change for finite-cycle spectra using permutation quotients, families of cycles, and arithmetic power homotopies, then passes to general bases by universal homeomorphism invariance and continuity.
Related
- Voevodsky motives and motives with modulus with finite coefficients in characteristic zero
- The zeroth stable homotopy groups of motivic spheres over the integers
- Pontryagin-Thom, π0 of the motivic sphere, and principal quadratic divisors
- The Arithmetic of Reducible Rank 2 Hypergeometric Motives
- Isotropic localizations of mixed initial motives
BibTeX
@misc{zhang2026voevodskys,
title = {{Voevodsky's motives and zeroth slice of motivic sphere}},
author = {Bo Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34898},
eprint = {2610.34898},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34898},
}
RIS
TY - UNPB
TI - Voevodsky's motives and zeroth slice of motivic sphere
AU - Bo Zhang
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.34898
UR - https://arxiv.org/abs/2610.34898
AB - Let S be a noetherian scheme of finite Krull dimension and let R be a commutative ring in which every positive residue characteristic of S is invertible. We prove that Voevodsky's stable Nisnevich category of motives with R-coefficients is canonically equivalent to modules over the zeroth slice of the motivic sphere with derived R-coefficients. The comparison identifies the corresponding commutative coefficient algebras and is compatible with arbitrary pullback. The theorem applies to singular and nonreduced schemes, including mixed-characteristic bases, and requires no excellence or finite-type hypothesis. Its special cases include rational coefficients over arbitrary bases, arbitrary coefficients over ℚ-schemes, and ℤ[1/p]-algebra coefficients over ℤ_(p)-schemes. The proof establishes base change for finite-cycle spectra using permutation quotients, families of cycles, and arithmetic power homotopies, then passes to general bases by universal homeomorphism invariance and continuity.
ER -