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+++ title = “The Motivic Adams Spectral Sequence” section = “mathematics” +++

The Motivic Adams Spectral Sequence

The motivic Adams spectral sequence is the Adams spectral sequence in motivic homotopy theory over $\operatorname{Spec} \mathbb{Z}$. It is trigraded by stem, filtration, and motivic weight, and converges to the motivic stable homotopy groups.

Setup

The motivic Steenrod algebra $A_{\mathrm{mot}}$ over $\mathbb{F}2$ is generated by Steenrod squares $\operatorname{Sq}^i$ subject to the Adem relations, but now carries a weight grading. In htpy’s conventions, a degree is a pair $(t, w)$ where $t$ is the internal degree and $w$ is the weight. The element $\tau \in \pi{0,-1}$ plays a central role: it has stem 0, filtration 0, and weight $-1$.

The motivic Adams spectral sequence takes the form

$$E_2^{s,t,w} = \operatorname{Ext}{A{\mathrm{mot}}}^{s,t,w}(\mathbb{F}2,; \mathbb{F}2) \implies \pi{t-s,w}(S^{0,0}{\mathrm{mot}})^\wedge_2$$

htpy computes the $E_2$ page by constructing a minimal free resolution of $\mathbb{F}2$ over $A{\mathrm{mot}}$, keeping track of the weight grading at each step.

Weight grading

Each generator of Ext has a well-defined weight $w$. The key constraint is:

Differentials preserve weight. The motivic Adams differential $d_r : E_r^{s,t,w} \to E_r^{s+r, t+r-1, w}$ has weight 0.

This is the basis of the weight degree argument (Pass 1.1 in the deduction engine): if the source and target of a potential $d_r$ have no generators at a common weight, then $d_r = 0$.

$\tau$-torsion

The element $\tau$ acts on $\operatorname{Ext}{A{\mathrm{mot}}}$ by shifting weight by $-1$. An element $x$ is $\tau$-torsion if $\tau^k \cdot x = 0$ for some $k \geq 1$. The minimal such $k$ is the $\tau$-torsion order of $x$.

$\tau$-torsion is a structural invariant of the $E_2$ page — it does not change across pages of the spectral sequence. The $\tau$-torsion orders constrain differentials in several ways:

  1. $\tau$-torsion obstruction (Pass 1.2): If $x$ has torsion order $k$, then $d_r(x)$ must land in an element with compatible torsion order $\leq k$. If no such target exists, $d_r(x) = 0$.

  2. $\tau$-survival (Pass 1.3): If $x$ has torsion order $k$, then $d_r(x) = 0$ for all $r \leq k$. This follows from Chua’s Lemma 12.4: $\tau^k$-torsion classes lie in $\operatorname{im}(d_{k+1})$, so they survive until page $E_{k+1}$.

  3. $\tau$-target forcing (Pass 3): If $x$ has exact torsion order $k$, then $x \in \operatorname{im}(d_{k+1})$. When the source of $d_{k+1}$ is 1-dimensional and the target is uniquely determined, the differential is forced.

See Deduction Engine: Mathematical Justification for the precise statements.

Computing $\tau$-torsion

htpy computes $\tau$-torsion orders from the long exact sequence in Ext associated to the cofiber sequence

$$0 \to \Sigma^{0,1}\mathbb{F}_2 \xrightarrow{i} C\tau \xrightarrow{\pi} \mathbb{F}_2 \to 0$$

The connecting homomorphism $\delta : \operatorname{Ext}^s(\mathbb{F}_2) \to \operatorname{Ext}^{s+1}(\Sigma^{0,1}\mathbb{F}_2)$ detects $\tau$-torsion:

$$x \in \operatorname{im}(\delta_{s-1}) \iff x \text{ is } \tau\text{-torsion}$$

For higher torsion orders, htpy uses the iterated LES: the short exact sequence $0 \to \Sigma^{0,1}C_{k-1} \to C_k \to \mathbb{F}_2 \to 0$ where $C_k = \mathbb{F}_2[\tau]/(\tau^k)$ gives a connecting homomorphism $\delta_k$ whose image detects ${x : \tau^{k-1} \cdot x = 0}$. The exact torsion order is $\min{m : x \in \operatorname{im}(\delta_m)}$.

See $C\tau$ Bridge for the full story.

The archive module S_2_mot

The archive module S_2_mot contains:

  • $E_2$ page through stem 110, filtration 55
  • $\tau$-torsion orders (computed via iterated LES for $k \leq 4$, supplemented by Isaksen’s CSV data at higher stems)
  • Differentials $d_2$ through $d_6$ from Isaksen’s tables
  • Products with $h_0, h_1, h_2, h_3$
  • Weight annotations per generator

In the viewer, $\tau$-torsion generators are displayed with hollow circles and their torsion order is shown in the detail panel. The weight filter dropdown allows viewing a single weight slice at a time.