Filtrations in ℂ-motivic stable homotopy theory

2026/08/05 by Konstantin Emming
Mathematics · #math.AT #math.AG #msc:14F42 #msc:55P42 #msc:55T99

paper · pdf

43 pages. Comments welcome!

arxiv created 2026/08/05 · arxiv updated 2026/08/06

Abstract

We study the effective, connective, and very effective filtrations in the ℂ-motivic, 2-complete, cellular, stable homotopy category. We do so by using the filtered spectrum model for this category due to Gheorghe-Isaksen-Krause-Ricka, and in particular the motivic analogue functor Γ_⋆. Then we can express the covers making up the respective filtrations of a nice motivic analogue Γ_⋆(X) via filtered spectra, and use these to compute the slices. Applying this in the case of X being the sphere spectrum, MU, ku, or an Eilenberg-MacLane spectrum recovers a number of conjectures due to Voevodsky. Applying it to ko recovers a computation of Ananyevskiy-Röndigs-Østvær. We can also apply it to tmf and compute the effective slices of the motivic modular forms spectrum mmf. We also study the effective slice spectral sequence for Γ_⋆(X), which turns out to contain the same information as the classical Adams-Novikov spectral sequence for X.

Citations

• An introduction to filtered and synthetic spectra
• On the Last Kervaire Invariant Problem
• Galois reconstruction of Artin-Tate ℝ-motivic spectra
• Stable homotopy groups of spheres: From dimension 0 to 90
• Voevodsky's slice conjectures via Hilbert schemes
• C-motivic modular forms
• On equivariant and motivic slices
• Synthetic spectra and the cellular motivic category
• Synthetic spectra and the cellular motivic category
• On very effective hermitian K-theory
• The Generalized Slices of Hermitian K-Theory
• The first stable homotopy groups of motivic spheres
• Stable stems
• Stable Stems
• The Adams-Novikov spectral sequence and Voevodsky's slice tower
• From algebraic cobordism to motivic cohomology
• A comparison of motivic and classical homotopy theories
• Motivic slices and colored operads
• Motivic twisted K-theory
• Relations between slices and quotients of the algebraic cobordism spectrum
• Multiplicative Properties of the Slice Filtration
• Motivic Landweber Exactness
• On motivic cohomology with Z/l coefficients
• Slices of motivic Landweber spectra
• Motivic cohomology with 𝐙 / 2 -coefficients
• Computation of the homotopy of the spectrum tmf
• Motivic cell structures
• On the zero slice of the sphere spectrum