The Motivic Adams Spectral Sequence and the Stable Stems¶

Job: develop the motivic Adams spectral sequence (with $\mathbf{H}\mathbb{F}_\ell$ / $\mathrm{MGL}$), the motivic Steenrod algebra, the motivic Adams–Novikov, the low-dimensional Milnor–Witt stems, and the $\mathbb{R}$-motivic / $C_2$-equivariant connection (Hill–Hopkins–Ravenel, Kervaire invariant) — with the motivic Adams computation of $\pi_{n,0}\mathbb{1}$ over $\mathbb{C}$ as the worked example.

Core¶

The motivic Adams spectral sequence¶

For a motivic spectrum $E\in\mathrm{SH}(k)$ and a motivic ring spectrum $R$ (with $R_*R$ flat over $R_*$), the motivic Adams spectral sequence $$ E_2^{s,t,w}=\mathrm{Ext}^{s,t,w}{R}(R_,R_*\wedge E)\Rightarrow \pi_{t-s,w}E\hat{}_R $$ (the $E_2$-page is Ext over the coefficient ring $R_*$, the bigrading $(t-s,w)$ the motivic stem with weight, converging to the $R$-completion $\widehat{\pi_{t-s,w}E}_R$).

With $R=\mathbf{H}\mathbb{F}_\ell$ (the mod-$\ell$ motivic Eilenberg–MacLane), the $E_2$-page is $$ E_2=\mathrm{Ext}{\mathcal{A}^{\mathrm{mot}}\ell}(\mathbb{F}\ell,\mathbb{F}\ell) $$ over the motivic Steenrod algebra $\mathcal{A}^{\mathrm{mot}}_\ell$, converging to the $\ell$-complete motivic stable stems $\widehat{\pi_{*,*}\mathbb{1}}_\ell$.

Spectral sequence	$E_2$-page	Converges to
Motivic Adams ($\mathbf{H}\mathbb{F}_\ell$)	$\mathrm{Ext}_{\mathcal{A}^{\mathrm{mot}}_\ell}(\mathbb{F}_\ell,\mathbb{F}_\ell)$	$\ell$-complete $\pi_{,}\mathbb{1}$
Motivic Adams–Novikov ($\mathrm{MGL}$)	$\mathrm{Ext}_{\mathrm{MGL}_{,}}(\mathrm{MGL}_{,},\mathrm{MGL}_{,})$	$\ell$-complete $\pi_{,}\mathbb{1}$
Slice spectral sequence (sphere)	$H^{p-q}(s_q\mathbb{1})$	$\pi_{p+q}\mathbb{1}$ (the geometric version)
Classical Adams	$\mathrm{Ext}_{\mathcal{A}_\ell}(\mathbb{F}_\ell,\mathbb{F}_\ell)$	$\ell$-complete $\pi_*^{\mathrm{top}}\mathbb{S}$
Betti realization	collapse weight	$\pi_n^{\mathrm{top}}\mathbb{S}$ from $\pi_{n,0}^{\mathrm{mot}}\mathbb{1}$

The motivic Steenrod algebra¶

The Steenrod operations $Sq^i$ (at $\ell=2$) and the reduced power operations $P^i$ (at odd $\ell$) act on the motivic cohomology $H^{*,*}(X,\mathbb{F}_\ell)$ (the Voevodsky construction, the motivic analog of the topological Steenrod algebra). The motivic Steenrod algebra $\mathcal{A}^{\mathrm{mot}}_\ell$ over $\mathbb{F}_\ell$ has:

the $Sq^i$ (and $P^i$) with the Adem relations (the motivic Adem relations, the same as the classical with the weight grading);
the Milnor primitives $Q_i$ (the Milnor $Q_i$ operations, related to the Milnor $K$-theory: $Q_0=\beta Sq^0$ the Bockstein, $Q_1=[Q_0,Sq^{2^1}]$ etc., with $Q_i$ detecting the $\eta$-family in the motivic stems);
the weight grading: $Sq^i$ has bidegree $(i, \lfloor i/2\rfloor)$ in motivic cohomology (the motivic Steenrod algebra is bigraded, the classical is singly graded).

Motivic $\mathcal{A}^{\mathrm{mot}}_\ell$	Classical $\mathcal{A}_\ell$
bigraded (degree, weight)	singly graded (degree)
$Sq^i$ of bidegree $(i,\lfloor i/2\rfloor)$	$Sq^i$ of degree $i$
$Q_0$ related to $\mathbb{G}_m$ / $\eta$	$Q_0$ the Bockstein
Adem relations with weight	Adem relations
depends on char$(k)$ and $\ell$	characteristic-independent

The motivic Steenrod algebra depends on the characteristic of $k$ and the prime $\ell$ (the relation to the Milnor $K$-theory $K_n^M(k)$ at $\ell$-torsion, the norm residue theorem — Voevodsky–Rost — identifying $H^{n,n}(X,\mathbb{F}_\ell)\cong H^n_{\mathrm{\acute{e}t}}(X,\mathbb{F}_\ell(n))$ in the appropriate range).

The motivic Adams–Novikov spectral sequence¶

Using the algebraic cobordism $\mathrm{MGL}$ (the universal oriented theory — see algebraic cobordism) as the Adams–Novikov input, the $E_2$-page $$ E_2=\mathrm{Ext}{\mathrm{MGL}{,}}(\mathrm{MGL}{,},\mathrm{MGL}{,}) $$ converges to the $\ell$-complete stable stems. The relation to the slice spectral sequence: the slice spectral sequence is the "geometric" version (the slices are motivic cohomology with the Lazard ring coefficients), the Adams–Novikov is the "algebraic" version — both converge to the same stems, with the Adams–Novikov the more computable (the Ext over the Lazard ring / the Hopf algebroid structure).

The Hopkins–Morel–Ravenel computation: the $\mathrm{MGL}$-based Novikov (the motivic analog of the Ravenel computation of the classical Adams–Novikov) gives the $\ell$-complete stems in a range, with the chromatic filtration (the $v_n$-periodic layers) the motivic analog of the classical chromatic picture.

The stable stems and the Milnor–Witt connection¶

The low-dimensional motivic stable stems $\pi_{p,q}\mathbb{1}$ (over $k=\mathbb{C},\mathbb{R}$):

Stem	Value over $k$	Identification
$\pi_{0,0}\mathbb{1}$	$\mathrm{GW}(k)$	Morel — the Grothendieck–Witt ring
$\pi_{1,1}\mathbb{1}$	$k^\times$ (via $\mathbf{K}^{\mathrm{MW}}_1$)	the Hopf map $\eta$ family (the algebraic Hopf)
$\pi_{n,n}\mathbb{1}$	$\mathbf{K}^{\mathrm{MW}}_n(k)$	Morel — the "diagonal" / Milnor–Witt stems
$\pi_{1,0}\mathbb{1}$	$\mathbb{Z}/2\{\eta_{\mathrm{top}}\}$ (over $\mathbb{C}$)	the "topological" $\eta$ (the Betti image of the classical Hopf)
$\pi_{n,0}\mathbb{1}$	$\pi_n^{\mathrm{top}}\mathbb{S}$ in a range	the Betti realization isomorphism (the "motivic = topological" range)

The Betti realization $\mathrm{SH}(\mathbb{C})\to\mathrm{Sp}$ (sending $\mathbb{P}^1\mapsto S^2$, collapsing the weight) induces $\pi_{n,0}^{\mathrm{mot}}\mathbb{1}\to\pi_n^{\mathrm{top}}\mathbb{S}$ which is an isomorphism in a range (the "motivic = topological" range, the stable range of the Betti realization — Isaksen and collaborators), giving the motivic refinement of the classical stable stems with the extra weight information.

The $\mathbb{R}$-motivic case and the $C_2$-equivariant connection¶

For $k=\mathbb{R}$, the real realization $\mathrm{SH}(\mathbb{R})\to\mathrm{SH}^{C_2}$ (the $C_2$-equivariant stable homotopy, the genuine $C_2$-spectra, via the Galois action of $C_2=\mathrm{Gal}(\mathbb{C}/\mathbb{R})$ on the Betti realization) connects the $\mathbb{R}$-motivic Adams to the $C_2$-equivariant Adams.

Theorem (Hill–Hopkins–Ravenel). The Kervaire invariant one problem (the existence of framed manifolds of Kervaire invariant one in dimensions $2^k-2$) is resolved via the $C_2$-equivariant / $\mathbb{R}$-motivic Adams spectral sequence: the Kervaire invariant one elements exist only in dimensions $2,6,14,30,62$ (and not in $126$ and higher). The $\mathbb{R}$-motivic Adams spectral sequence (the Isaksen–Wang–Xu computation) computes the $\mathrm{RO}(C_2)$-graded $C_2$-equivariant stems, detecting the Kervaire-invariant-one classes.

$\mathbb{R}$-motivic	$C_2$-equivariant
$\mathrm{SH}(\mathbb{R})$	$\mathrm{SH}^{C_2}$ (genuine $C_2$-spectra)
$\pi_{p,q}^{\mathbb{R}}\mathbb{1}$	$\mathrm{RO}(C_2)$-graded equivariant stems
$\mathbb{R}$-motivic Adams	$C_2$-equivariant Adams (Hill–Hopkins–Ravenel)
Kervaire-invariant-one resolution	the $126$-stem and above (no Kervaire-one)

See it¶

Motivic Adams spectral sequence computation of $\pi_{n,0}\mathbb{1}$ over $\mathbb{C}$ in low stem. The $E_2$-page $\mathrm{Ext}_{\mathcal{A}^{\mathrm{mot}}_2}(\mathbb{F}_2,\mathbb{F}_2)$ over the mod-2 motivic Steenrod algebra (the bigraded Ext) gives the first few stems:

$\pi_{0,0}\mathbb{1}=\mathbb{Z}$ (the Adams summand, $\mathrm{GW}(\mathbb{C})=\mathbb{Z}$ via the rank map; the $0$-stem).
$\pi_{1,1}\mathbb{1}=\mathbb{Z}/2\{\eta\}$ (the algebraic Hopf map $\eta$, the diagonal stem — the Hopf map $S^{2,1}\to S^{1,0}$ detecting the unit $-1\in\mathbb{C}^\times$ via Morel's identification $\pi_{1,1}\cong k^\times/(k^\times)^2$ reduced mod squares).
$\pi_{1,0}\mathbb{1}=\mathbb{Z}/2\{\eta_{\mathrm{top}}\}$ (the "topological" $\eta$, the Betti image of the classical Hopf map $\eta\in\pi_1^{\mathrm{top}}\mathbb{S}$ — the weightless stem).
$\pi_{2,1},\pi_{2,2},\pi_{2,0}$: the $\eta^2$, $\nu$-families, the higher Hopf maps.

The comparison with the classical stable stems: the Betti realization sends $\pi_{n,0}^{\mathrm{mot}}\mathbb{1}\to\pi_n^{\mathrm{top}}\mathbb{S}$ the classical stem, isomorphically in a range (the "motivic = topological" range). The key refinement: the motivic Adams spectral sequence resolves the classical stable stems with the extra weight information — the classical $\eta\in\pi_1^{\mathrm{top}}\mathbb{S}$ splits motivically into $\eta\in\pi_{1,1}$ (the algebraic Hopf) and $\eta_{\mathrm{top}}\in\pi_{1,0}$ (the topological Hopf), giving a finer computation. The motivic Adams $E_2$-page has the algebraic $\eta$ detected by $Q_0$ (the Bockstein, the Milnor primitive related to $\mathbb{G}_m$) and the topological $\eta$ detected by the classical $h_1$ Adams class, separating the two via the weight grading.

Check¶

Why does the Betti realization $\pi_{n,0}^{\mathrm{mot}}\mathbb{1}\to\pi_n^{\mathrm{top}}\mathbb{S}$ fail to be an isomorphism in all degrees, and what is the obstruction?

The Betti realization is an isomorphism in a range (the "stable range", roughly $n\le$ a bound depending on the connectivity, Isaksen) but fails in high degree because the motivic stems have extra weight-graded classes (the $\eta$-multiplication towers, the Milnor–Witt $K$-theory contributions in $\pi_{n,n}$) that do not survive the Betti realization (the weight is collapsed). The obstruction is the weight: $\pi_{p,q}^{\mathrm{mot}}$ with $q>0$ (the "weighted" stems) have no classical counterpart (the Betti realization sends $q\mapsto 0$), so the full motivic stems are richer than the classical — the realization is surjective (every classical stem has a motivic lift) but not injective in general (motivic stems can have extra weight information that the realization kills).

Show that the motivic $\eta\in\pi_{1,1}\mathbb{1}$ is detected by the Milnor primitive $Q_0\in\mathcal{A}^{\mathrm{mot}}_2$, and identify the relation to $\mathbb{G}_m$.

The motivic $\eta:S^{2,1}\to S^{1,0}$ is the Hopf map represented by $-1\in k^\times$ (the map $\mathbb{A}^2\setminus 0\to\mathbb{P}^1$, the clutching function $-1$). In the Adams $E_2$-page, the class $h_0$ (the Bockstein $Q_0=\beta Sq^0$) detects the $\eta$-multiplication / the degree-1 element. The Milnor primitive $Q_0$ is the motivic Bockstein (the connecting homomorphism $H^{*,*}(-,\mathbb{F}_2)\to H^{*+1,*}(-,\mathbb{F}_2)$, the boundary of the coefficient sequence $0\to\mathbb{F}_2\to\mathbb{Z}/4\to\mathbb{F}_2\to 0$), and it is related to $\mathbb{G}_m$ via the identification $H^{1,1}(\mathrm{Spec}\,k,\mathbb{F}_2)\cong k^\times/(k^\times)^2$ (the norm residue / the Bloch–Kato theorem at $\ell=2$), so $Q_0$ on the unit class gives the class of $-1\in k^\times/(k^\times)^2$, which is exactly the Hopf $\eta$. So $Q_0$ detects $\eta$ via the $\mathbb{G}_m$ / Milnor $K$-theory connection.

Pitfalls¶

The motivic Adams spectral sequence converges to the $\ell$-complete stems (the completion $\widehat{\pi_{*,*}\mathbb{1}}_\ell$), not the integral stems — the $\ell$-adic completion is essential (the Adams spectral sequence computes the $\ell$-completion, the inverse limit over $\ell^m$).
The motivic Steenrod algebra depends on the characteristic (the $p$ vs $\ell$, the relation to the Milnor $K$-theory at $\ell$-torsion) — for $\ell=$ char$(k)$ the Steenrod algebra is different (the $V$-operations, the Milnor $Q_i$ related to the $\ell$-adic $K$-theory).
The Betti realization $\pi_{n,0}^{\mathrm{mot}}\to\pi_n^{\mathrm{top}}$ is an isomorphism in a range (not in all degrees) — the "motivic = topological" range is bounded; in high degree the motivic stems have extra weight information.
The $\mathbb{R}$-motivic / $C_2$-equivariant correspondence is the Hill–Hopkins–Ravenel framework (the Kervaire invariant is resolved via the $\mathbb{R}$-motivic Adams, a non-trivial theorem, not a formal correspondence) — the real realization $\mathrm{SH}(\mathbb{R})\to\mathrm{SH}^{C_2}$ is an equivalence on the $\ell$-complete subcategories in a range (the equivalence of the $\mathbb{R}$-motivic and the $C_2$-equivariant stable homotopy, the Hu–Kriz–Ormsby theorem, refined by Hill–Hopkins–Ravenel).
The motivic Adams / Adams–Novikov comparison: the two spectral sequences converge to the same $\ell$-complete stems but the $E_2$-pages are different (the Adams $E_2$ is Ext over the Steenrod algebra, the Adams–Novikov $E_2$ is Ext over the Lazard ring); the comparison is via the chromatic / the slice spectral sequence.
The Kervaire invariant one resolution (Hill–Hopkins–Ravenel) is via the $C_2$-equivariant Adams, not directly the $\mathbb{R}$-motivic Adams — the $\mathbb{R}$-motivic computation (Isaksen–Wang–Xu) is the computational realization of the HHR framework, not the HHR theorem itself.

Refs¶

Morel, $\mathbb{A}^1$-Algebraic Topology over a Field (LNM 2012) — $\pi_{n,n}\mathbb{1}=\mathbf{K}^{\mathrm{MW}}_n$
Voevodsky, "Reduced powers and the motivic Steenrod algebra" (2003); the norm residue / Bloch–Kato theorem (Voevodsky–Rost, 2011)
Hill–Hopkins–Ravenel, "On the nonexistence of elements of Kervaire invariant one" (Annals 2016)
Isaksen–Wang–Xu, "More stable stems" (2023); Isaksen, "Stable stems" (2014–); Hu–Kriz–Ormsby, "The $\mathbb{R}$-motivic and $C_2$-equivariant Adams spectral sequences"
Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres (the classical Adams–Novikov, the motivic analog)

Next¶

MH index · Foundations · Six Functors & Base Change · Slice Filtration · Algebraic Cobordism · Algebraic topology · K-theory · SAG / chromatic — the Morava $E$-theory / the chromatic filtration as the spectral-geometric framework for the stable stems

Spectral sequence	\(E_2\)-page	Converges to
Motivic Adams (\(\mathbf{H}\mathbb{F}_\ell\))	\(\mathrm{Ext}_{\mathcal{A}^{\mathrm{mot}}_\ell}(\mathbb{F}_\ell,\mathbb{F}_\ell)\)	\(\ell\)-complete \(\pi_{,}\mathbb{1}\)
Motivic Adams–Novikov (\(\mathrm{MGL}\))	\(\mathrm{Ext}_{\mathrm{MGL}_{,}}(\mathrm{MGL}_{,},\mathrm{MGL}_{,})\)	\(\ell\)-complete \(\pi_{,}\mathbb{1}\)
Slice spectral sequence (sphere)	\(H^{p-q}(s_q\mathbb{1})\)	\(\pi_{p+q}\mathbb{1}\) (the geometric version)
Classical Adams	\(\mathrm{Ext}_{\mathcal{A}_\ell}(\mathbb{F}_\ell,\mathbb{F}_\ell)\)	\(\ell\)-complete \(\pi_*^{\mathrm{top}}\mathbb{S}\)
Betti realization	collapse weight	\(\pi_n^{\mathrm{top}}\mathbb{S}\) from \(\pi_{n,0}^{\mathrm{mot}}\mathbb{1}\)

Motivic \(\mathcal{A}^{\mathrm{mot}}_\ell\)	Classical \(\mathcal{A}_\ell\)
bigraded (degree, weight)	singly graded (degree)
\(Sq^i\) of bidegree \((i,\lfloor i/2\rfloor)\)	\(Sq^i\) of degree \(i\)
\(Q_0\) related to \(\mathbb{G}_m\) / \(\eta\)	\(Q_0\) the Bockstein
Adem relations with weight	Adem relations
depends on char\((k)\) and \(\ell\)	characteristic-independent

Stem	Value over \(k\)	Identification
\(\pi_{0,0}\mathbb{1}\)	\(\mathrm{GW}(k)\)	Morel — the Grothendieck–Witt ring
\(\pi_{1,1}\mathbb{1}\)	\(k^\times\) (via \(\mathbf{K}^{\mathrm{MW}}_1\))	the Hopf map \(\eta\) family (the algebraic Hopf)
\(\pi_{n,n}\mathbb{1}\)	\(\mathbf{K}^{\mathrm{MW}}_n(k)\)	Morel — the "diagonal" / Milnor–Witt stems
\(\pi_{1,0}\mathbb{1}\)	\(\mathbb{Z}/2\{\eta_{\mathrm{top}}\}\) (over \(\mathbb{C}\))	the "topological" \(\eta\) (the Betti image of the classical Hopf)
\(\pi_{n,0}\mathbb{1}\)	\(\pi_n^{\mathrm{top}}\mathbb{S}\) in a range	the Betti realization isomorphism (the "motivic = topological" range)

\(\mathbb{R}\)-motivic	\(C_2\)-equivariant
\(\mathrm{SH}(\mathbb{R})\)	\(\mathrm{SH}^{C_2}\) (genuine \(C_2\)-spectra)
\(\pi_{p,q}^{\mathbb{R}}\mathbb{1}\)	\(\mathrm{RO}(C_2)\)-graded equivariant stems
\(\mathbb{R}\)-motivic Adams	\(C_2\)-equivariant Adams (Hill–Hopkins–Ravenel)
Kervaire-invariant-one resolution	the \(126\)-stem and above (no Kervaire-one)

The Motivic Adams Spectral Sequence and the Stable Stems¶

Core¶