Generating hypotheses along the motivic deformation
Abstract
At every prime p, we disprove the algebraic generating hypothesis for compact objects in Hovey's stable category of BP*BP-comodules and its local analog at every positive height, answering questions of Barthel and Heard. In both settings, for every r,n≥1, we construct a ghost whose first n composition powers are all nonzero and have exact additive order pr. The same construction applies to the stable category of E*E-comodules for even p-local Landweber exact theories E that are not rational. Through the motivic deformation of Gheorghe--Wang--Xu, these examples give Cτ-linear counterexamples to the cellular ℂ-motivic generating hypothesis, with the same order and composition properties. We also construct nonzero non-Cτ-linear ghosts between Cτ-modules and a nonzero ghost on a motivic spectrum with noncontractible Betti realization.
Related
- Equivariant generating hypotheses for finite groups
- Schematic Functorialities of Birational Motivic Homotopy Categories
- On the internal homotopy theory of motivic categories
- ℂ-Motivic Eilenberg-Mac Lane Spectra as MGL-Motivic Thom Spectra and Normed 𝔾m-Hochschild Homology
- Motivic invariants of birational hyper-Kähler manifolds
BibTeX
@misc{ma2026generating,
title = {{Generating hypotheses along the motivic deformation}},
author = {Sihao Ma and Zhouli Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15253},
eprint = {2609.15253},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15253},
}
RIS
TY - UNPB
TI - Generating hypotheses along the motivic deformation
AU - Sihao Ma
AU - Zhouli Xu
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.15253
UR - https://arxiv.org/abs/2609.15253
AB - At every prime p, we disprove the algebraic generating hypothesis for compact objects in Hovey's stable category of BP_*BP-comodules and its local analog at every positive height, answering questions of Barthel and Heard. In both settings, for every r,n≥1, we construct a ghost whose first n composition powers are all nonzero and have exact additive order p^r. The same construction applies to the stable category of E_*E-comodules for even p-local Landweber exact theories E that are not rational. Through the motivic deformation of Gheorghe--Wang--Xu, these examples give Cτ-linear counterexamples to the cellular ℂ-motivic generating hypothesis, with the same order and composition properties. We also construct nonzero non-Cτ-linear ghosts between Cτ-modules and a nonzero ghost on a motivic spectrum with noncontractible Betti realization.
ER -