SLICE FILTRATION TOWERS

I. Slice Filtration

Voevodsky's slice filtration is the motivic analog of the Postnikov tower. For a motivic spectrum E ∈ SH(k), the slice tower:

... → f_{n+1} E → f_n E → f_{n-1} E → ... → E

where f_n E is the n-effective cover (connective cover in the slice direction). The n-th slice is the fiber:

s_n E = fib(f_n E → f_{n+1} E)

II. Effective Motivic Spectra

A motivic spectrum E is effective (n-effective) if E ∈ SH^{eff}(k) = Σ^{∞}_{P^1} H(k). The effective subcategory is generated by suspension spectra of smooth schemes. The slice tower converges:

E ≅ holim_n f_n E for E effective

The slices s_n(E) are pure of weight n: they are modules over the n-th power of the Tate motive.

III. Slice of KGL

The algebraic K-theory spectrum KGL has slices computed by Voevodsky:

s_n(KGL) ≅ Σ^{2n,n} HZ (the shifted motivic Eilenberg-MacLane)

This recovers the motivic Atiyah-Hirzebruch spectral sequence for K-theory from motivic cohomology.

IV. Cellular Approximation

A motivic spectrum is cellular if it is built from Σ^{p,q} S^0 by colimits. Cellular spectra admit a CW-structure in the motivic sense:

Cell(SH(k)) ⊆ SH(k), generated by bigraded spheres S^{p,q}

Many important spectra (KGL, MGL, HZ) are cellular. Cellular approximation provides computable filtrations.

◆ SLICE FILTRATION TOWERS

I. Slice Filtration

II. Effective Motivic Spectra

III. Slice of KGL

IV. Cellular Approximation