Motivic Telescope Conjecture in telescopic form

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Let XX be a finite motivic spectrum of type (m,n)(m,n) with a non-nilpotent self-map v:Σ−r,−sX→Xv:\Sigma^{-r,-s}X\to X of type (m,n)(m,n). Write ⟨E⟩\langle E\rangle for the Bousfield class of a motivic spectrum EE, and v−1Xv^{-1}X for the telescope of vv. Motivic Telescope Conjecture. The Bousfield class ⟨v−1X⟩\langle v^{-1}X\rangle depends only on mm and nn, and

⟨v−1X⟩=⟨K(βij)⟩.\langle v^{-1}X\rangle=\langle K(\beta_{ij})\rangle.

This is the telescopic formulation of the motivic Telescope Conjecture; its relationship with the localization and smashing formulations depends on further conjectural smashing and localization properties.

References

Primary source

Dominic Leon Culver and J. D. Quigley, “kq-Resolutions I”, arXiv:1905.11952 (2020).

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