The Slice Filtration and the Motivic Postnikov / Chromatic Tower¶

Job: develop Voevodsky's slice filtration on $\mathrm{SH}(S)$ — the slice tower, the slice spectral sequence, the slices of the sphere and of $\mathrm{MGL}$ / $\mathrm{KGL}$, the effective / very effective $t$-structures, and the connection to the Milnor–Witt stems — with the slice spectral sequence for $\mathbb{1}$ and $\mathrm{MGL}$ over $\mathbb{C}$ as the worked example.

Core¶

The slice filtration (Voevodsky)¶

For $E\in\mathrm{SH}(S)$, the slice tower is a decreasing filtration $$ \cdots\to f_{q+1}E\to f_qE\to f_{q-1}E\to\cdots $$ by the slice functors $f_q$ (the localizing subcategories generated by $\Sigma^{2q,q}\mathrm{SH}^{\mathrm{eff}}(S)$, where $\mathrm{SH}^{\mathrm{eff}}(S)$ is the effective subcategory — the localizing subcategory generated by $\Sigma^\infty_{\mathbb{P}^1}X_+$ for $X\in\mathbf{Sm}/S$ without negative $\mathbb{P}^1$-suspension). The slices are the associated graded: $$ s_qE=f_qE/f_{q+1}E. $$ The slices $s_qE$ are $\mathbf{H}\mathbb{Z}$-modules (they lie in the heart of the homotopy $t$-structure on $\mathrm{SH}^{\mathrm{eff}}$, i.e. they are $\mathbb{A}^1$-invariant / effective motives — Voevodsky's theorem), and the slice tower is the motivic analog of the Postnikov tower / the chromatic tower in classical stable homotopy.

Slice filtration	Classical analog
slice tower $f_qE\to E$	Postnikov tower $P_n E\to E$ (connective cover)
slices $s_qE=f_q/f_{q+1}$	homotopy groups $\pi_n E=E_n$ / chromatic layers $L_n E$
effective $\mathrm{SH}^{\mathrm{eff}}$	connective spectra $\mathrm{Sp}_{\ge0}$
very effective $\mathrm{SH}^{\mathrm{veff}}$	$(-1)$-connective / constructive spectra
slice spectral sequence	Adams / Adams–Novikov / chromatic spectral sequence

The slice spectral sequence¶

The slice tower gives the slice spectral sequence $$ E^1_{p,q}=H^{p-q}(s_qE)\Rightarrow H^{p+q}(E), $$ a right-half-plane spectral sequence converging to the motivic homotopy / homology groups of $E$ (the $H^*(s_qE)$ are the motivic cohomology groups of the slice). For $E=\mathbb{1}$ (the sphere spectrum), the slice spectral sequence is the motivic Adams–Novikov spectral sequence analog, converging to the stable stems $\pi_{p,q}\mathbb{1}$.

Theorem (Hayashi convergence). The slice tower is convergent for connective $E$ (i.e. $E\in\mathrm{SH}^{\mathrm{eff}}$ or more generally $E$ with bounded-negative slices): the slice spectral sequence converges to $H^{p+q}(E)$. The convergence is not unconditional for arbitrary $E$ (the slice tower may fail to be exhaustive / separated for non-connective $E$).

The sphere spectrum and the Milnor–Witt stems¶

The slice computation of $\mathbb{1}$ (Voevodsky, Röndigs–Østvær): the zeroth slice is $$ s_0\mathbb{1}=\mathbf{H}\mathbb{Z}, $$ the motivic Eilenberg–MacLane spectrum (integral motivic cohomology). The higher slices $s_q\mathbb{1}$ are related to the Milnor–Witt $K$-theory sheaves and the stable stems. The Milnor–Witt $K$-theory $\mathbf{K}^{\mathrm{MW}}_n$ (Morel): the stable homotopy sheaves $$ \underline{\pi}_{n,n}\mathbb{1}\cong\mathbf{K}^{\mathrm{MW}}_n, $$ the "diagonal" stems (the zero-th slice / the first non-trivial stems), with $\mathbf{K}^{\mathrm{MW}}_0(k)=\mathrm{GW}(k)$ the Grothendieck–Witt ring.

Stem	Value / identification
$\pi_{0,0}\mathbb{1}(k)$	$\mathrm{GW}(k)$ (Morel) — the Grothendieck–Witt ring
$\underline{\pi}_{n,n}\mathbb{1}$	$\mathbf{K}^{\mathrm{MW}}_n$ (Milnor–Witt $K$-theory, Morel)
$s_0\mathbb{1}$	$\mathbf{H}\mathbb{Z}$ (Voevodsky)
$s_q\mathbb{1}$ for $q>0$	related to $\mathbf{H}\mathbb{Z}$-modules of $K^{\mathrm{MW}}$ and the stable stems
"Weightless" $\pi_{n,0}\mathbb{1}$	the Betti realization to $\pi_n^{\mathrm{top}}\mathbb{S}$ is an iso in a range

Algebraic cobordism $\mathrm{MGL}$ and the slice filtration¶

The algebraic cobordism spectrum $\mathrm{MGL}=\mathrm{MGL}_S$ (Voevodsky, the Thom spectrum of the tautological bundles over $\mathrm{BGL}_{\infty,S}$ — see algebraic cobordism) is the algebraic analog of $\mathrm{MU}$.

Theorem (Levine; Spitzweck; Hopkins–Morel). The slices of $\mathrm{MGL}$ are the motivic Eilenberg–MacLane spectra of the algebraic cobordism groups: $$ s_q\mathrm{MGL}\cong\bigvee_{i\ge0}\Sigma^{2q+i,q}\mathbf{H}\mathbb{Z}{x_i}, $$ where $x_i$ range over a basis of the Lazard ring $L$ in degree $i$ (the complex cobordism coefficients). In particular $s_0\mathrm{MGL}\cong\mathbf{H}\mathbb{Z}$ (the Hopkins–Morel theorem: the zeroth slice of $\mathrm{MGL}$ is the motivic cohomology). The slice spectral sequence for $\mathrm{MGL}$ converges to the algebraic cobordism groups $\mathrm{MGL}_{*,*}(S)$.

The Rost motive (the motive of a Pfister quadric) and the Hopkins–Morel theorem connect the slice of $\mathrm{MGL}$ to the Milnor $K$-theory and the norm residue theorem (Voevodsky–Rost): the slice filtration detects the Milnor $K$-theory layers (the $K$-theory / norm residue connection via the Hopf map $\eta$ and the $\mathbb{A}^1$-sphere).

Spectrum	Slices
$\mathbb{1}$ (sphere)	$s_0=\mathbf{H}\mathbb{Z}$; higher slices $=\mathbf{H}\mathbb{Z}$-modules of $K^{\mathrm{MW}}$ + stems
$\mathbf{H}\mathbb{Z}$ (motivic cohomology)	$s_0=\mathbf{H}\mathbb{Z}$; $s_q=0$ for $q>0$ (it is already a slice)
$\mathrm{KGL}$ (algebraic $K$-theory)	$s_q=\Sigma^{2q,q}\mathbf{H}\mathbb{Z}$ (the slices recover the motivic cohomology, the Atiyah–Hirzebruch / the $\gamma$-filtration)
$\mathrm{MGL}$ (algebraic cobordism)	$s_q=\bigvee_i\Sigma^{2q+i,q}\mathbf{H}\mathbb{Z}\{x_i\}$ (the Lazard ring, Levine / Hopkins–Morel)

The effective / very effective $t$-structures¶

Effective category $\mathrm{SH}^{\mathrm{eff}}(S)$: the localizing subcategory generated by $\Sigma^{2q,q}\mathrm{SH}^{\mathrm{eff}}$ for $q\ge0$ — the non-negative slices, the connective part. The inclusion $\mathrm{SH}^{\mathrm{eff}}\hookrightarrow\mathrm{SH}$ has a right adjoint (the effective cover).
Very effective $\mathrm{SH}^{\mathrm{veff}}(S)$: the non-negative homotopy $t$-structure (the cellular / constructive part), the subcategory where $\underline{\pi}_{p,q}$ vanishes for $p<q$.
Homotopy $t$-structure on $\mathrm{SH}^{\mathrm{eff}}$: the heart is the category of $\mathbb{A}^1$-invariant sheaves with transfers — Voevodsky's $\mathrm{DM}^{\mathrm{eff}}$ (the effective motives). The relation: $\mathrm{DM}^{\mathrm{eff}}\simeq$ heart of the homotopy $t$-structure on $\mathrm{SH}^{\mathrm{eff}}$ (after tensoring with $\mathbb{Q}$ or integrally with care).
Coniveau / slice spectral sequence: the categorical framework unifying the slice filtration and the $t$-structure — the slice spectral sequence is the coniveau spectral sequence of the $t$-structure.

See it¶

Slice spectral sequence for the sphere spectrum over $\mathbb{C}$. Over $k=\mathbb{C}$ (where the Betti realization $\mathrm{SH}(\mathbb{C})\to\mathrm{Sp}$ is available), the slices of $\mathbb{1}$ are: - $s_0\mathbb{1}=\mathbf{H}\mathbb{Z}$ (the zeroth slice, Voevodsky); - $s_1\mathbb{1}=0$ (or a low group, depending on the parity); - the higher slices $s_q\mathbb{1}$ are related to the Milnor $K$-theory $K_n^M(\mathbb{C})=\mathbb{Z}/2$ (for $n\ge1$) and the Adams–Novikov $E_2$-page.

The slice spectral sequence $$ E^1_{p,q}=H^{p-q}(\mathbb{C},s_q\mathbb{1})\Rightarrow\pi_{p+q}(\mathbb{1}) $$ converges to the stable stems $\pi_{p,q}(\mathbb{1})$ with the Betti realization (sending $\mathbb{P}^1\mapsto S^2$, collapsing the weight) recovering the classical Adams–Novikov spectral sequence for $\pi_n^{\mathrm{top}}\mathbb{S}$. The motivic refinement: the extra weight grading separates the Hopf map $\eta\in\pi_{1,1}\mathbb{1}$ (the algebraic Hopf, $\eta:1\mapsto$ the Hopf map $S^{2,1}\to S^{1,0}$) from the "topological" $\eta\in\pi_{1,0}\mathbb{1}$ (the Betti image of the classical Hopf map), giving a finer spectral sequence.

Contrast: $\mathrm{MGL}$ slice spectral sequence. The slices of $\mathrm{MGL}$ are the $\mathbf{H}\mathbb{Z}$-modules of the Lazard ring $L=\mathbb{Z}[x_1,x_2,\dots]$ ($|x_i|=2i$): $$ s_q\mathrm{MGL}=\bigvee_{i\ge0}\Sigma^{2q+i,q}\mathbf{H}\mathbb{Z}{x_i}, $$ so the slice spectral sequence $$ E^1_{p,q}=H^{p-q}(\mathbb{C},s_q\mathrm{MGL})\Rightarrow\mathrm{MGL}_{p+q}(\mathbb{C}) $$ has $E^1$-page the motivic cohomology $H^*(\mathbb{C},\mathbf{H}\mathbb{Z}\{x_i\})=\mathbb{Z}\{x_i\}$ (concentrated in degree 0 over a point), converging to the algebraic cobordism ring $\mathrm{MGL}_{*,*}(\mathrm{pt})=L$ (the Lazard ring) — the algebraic analog of the Ravenel computation of $\mathrm{MU}_*=L$ via the Adams–Novikov spectral sequence. This is the motivic / slice-theoretic computation of the algebraic cobordism ring.

Check¶

Why is $s_0\mathbb{1}=\mathbf{H}\mathbb{Z}$ rather than $\mathbb{1}$ itself, and what does this say about the slice filtration?

The slice filtration peels off the "lower-weight" pieces: $s_0\mathbb{1}$ is the zeroth slice (the part of weight 0), which by Voevodsky's theorem is the motivic Eilenberg–MacLane $\mathbf{H}\mathbb{Z}$ (the "linear" / additive part of the sphere). The sphere $\mathbb{1}$ itself is non-linear (it has the Milnor–Witt stems, the Hopf maps $\eta,\nu,\sigma$, etc.), and the slice filtration separates the linear part ($s_0=\mathbf{H}\mathbb{Z}$) from the higher non-linear parts (the higher slices). The identification $s_0\mathbb{1}=\mathbf{H}\mathbb{Z}$ is a deep theorem (Voevodsky), not formal — it says the "weight-zero" piece of the sphere is exactly motivic cohomology.

Show that the slice of $\mathrm{KGL}$ recovers the motivic cohomology, and identify the relation to the $\gamma$-filtration.

The slices of $\mathrm{KGL}$ are $s_q\mathrm{KGL}=\Sigma^{2q,q}\mathbf{H}\mathbb{Z}$ (the slice of the periodic $K$-theory is the shifted motivic cohomology). The slice spectral sequence $E^1_{p,q}=H^{p-q}(\Sigma^{2q,q}\mathbf{H}\mathbb{Z})=H^{p-q-2q,q}(\mathbf{H}\mathbb{Z})\Rightarrow K_{p+q}(X)$ recovers the Atiyah–Hirzebruch spectral sequence for $K$-theory, i.e. the $\gamma$-filtration: the $K$-theory is built from the motivic cohomology (= the $\gamma$-filtration graded pieces $K_n^{(q)}\cong H^{2q-n,q}(X,\mathbb{Z})$). This is the motivic version of the topological Atiyah–Hirzebruch spectral sequence $H^p(X,\mathbb{Z})\Rightarrow K^p(X)$.

Pitfalls¶

The slice filtration converges for connective $E$ (Hayashi convergence), not unconditionally — for non-connective $E$ (e.g. periodic spectra without the connective cover), the slice tower may fail to be exhaustive / separated. Do not apply the slice spectral sequence to arbitrary $E$ without the connectivity check.
The slice $s_0\mathbb{1}=\mathbf{H}\mathbb{Z}$ is a theorem (Voevodsky), the higher slices are not all computed — only partial results (Röndigs–Østvær, Levine) for $s_q\mathbb{1}$ with $q$ small. The full slice computation of $\mathbb{1}$ is open.
The slices are $\mathbf{H}\mathbb{Z}$-modules (the heart of the homotopy $t$-structure); this is the slice theorem (Voevodsky / Levine), not a formal consequence — it is the deep input making the slice spectral sequence computable.
The effective category $\mathrm{SH}^{\mathrm{eff}}$ is the connective part; the very effective $\mathrm{SH}^{\mathrm{veff}}$ is the non-negative homotopy $t$-structure — they are different (the very effective is more restrictive). The relation to $\mathrm{DM}^{\mathrm{eff}}$ (Voevodsky's motives) is via the heart of the homotopy $t$-structure on $\mathrm{SH}^{\mathrm{eff}}$, not the slice filtration directly.
The $\mathrm{KGL}$ slice recovers the $\gamma$-filtration; the $\mathrm{MGL}$ slice recovers the Lazard ring — these are theorems (Levine, Hopkins–Morel), not formal consequences of the slice definition.
The slice spectral sequence for $\mathrm{MGL}$ converges to the algebraic cobordism ring, but the differentials are non-trivial (the higher Hopf maps $\eta,\nu,\sigma$ act as differentials) — the slice spectral sequence is not a degenerate / collapse at $E^1$ in general.

Refs¶

Voevodsky, "Open problems in the motivic stable homotopy theory I" (1999); "A possible new approach to the motivic stable homotopy theory" (2002); "On the zero slice of the sphere spectrum" (doc. math. 2004)
Pelaez, "Multiplicative properties of the slice filtration" (Astérisque 2011); Levine, "Slices of the motivic sphere and the Atiyah–Hirzebruch spectral sequence" (J. Topol. 2012)
Röndigs–Østvær, "Slices of motivic Landweber exact theories" (Adv. Math. 2016); Spitzweck, "Slices of motivic Landweber exact theories" (2018)
Hopkins–Morel, "Slices of the motivic sphere and the Lazard ring" (unpublished); Hayashi, "On the slice filtration" (J. Topol. 2018)

Next¶

MH index · Foundations · Six Functors & Base Change · Motivic Adams & Stable Stems · Algebraic Cobordism · K-theory · Algebraic topology

Slice filtration	Classical analog
slice tower \(f_qE\to E\)	Postnikov tower \(P_n E\to E\) (connective cover)
slices \(s_qE=f_q/f_{q+1}\)	homotopy groups \(\pi_n E=E_n\) / chromatic layers \(L_n E\)
effective \(\mathrm{SH}^{\mathrm{eff}}\)	connective spectra \(\mathrm{Sp}_{\ge0}\)
very effective \(\mathrm{SH}^{\mathrm{veff}}\)	\((-1)\)-connective / constructive spectra
slice spectral sequence	Adams / Adams–Novikov / chromatic spectral sequence

Stem	Value / identification
\(\pi_{0,0}\mathbb{1}(k)\)	\(\mathrm{GW}(k)\) (Morel) — the Grothendieck–Witt ring
\(\underline{\pi}_{n,n}\mathbb{1}\)	\(\mathbf{K}^{\mathrm{MW}}_n\) (Milnor–Witt \(K\)-theory, Morel)
\(s_0\mathbb{1}\)	\(\mathbf{H}\mathbb{Z}\) (Voevodsky)
\(s_q\mathbb{1}\) for \(q>0\)	related to \(\mathbf{H}\mathbb{Z}\)-modules of \(K^{\mathrm{MW}}\) and the stable stems
"Weightless" \(\pi_{n,0}\mathbb{1}\)	the Betti realization to \(\pi_n^{\mathrm{top}}\mathbb{S}\) is an iso in a range

Spectrum	Slices
\(\mathbb{1}\) (sphere)	\(s_0=\mathbf{H}\mathbb{Z}\); higher slices \(=\mathbf{H}\mathbb{Z}\)-modules of \(K^{\mathrm{MW}}\) + stems
\(\mathbf{H}\mathbb{Z}\) (motivic cohomology)	\(s_0=\mathbf{H}\mathbb{Z}\); \(s_q=0\) for \(q>0\) (it is already a slice)
\(\mathrm{KGL}\) (algebraic \(K\)-theory)	\(s_q=\Sigma^{2q,q}\mathbf{H}\mathbb{Z}\) (the slices recover the motivic cohomology, the Atiyah–Hirzebruch / the \(\gamma\)-filtration)
\(\mathrm{MGL}\) (algebraic cobordism)	\(s_q=\bigvee_i\Sigma^{2q+i,q}\mathbf{H}\mathbb{Z}\{x_i\}\) (the Lazard ring, Levine / Hopkins–Morel)