+++ title = “The Adams Spectral Sequence” section = “mathematics” +++
The Adams Spectral Sequence
The Adams spectral sequence is the primary computational tool in stable homotopy theory. For a spectrum $X$, it takes the form
$$E_2^{s,t} = \operatorname{Ext}_A^{s,t}(H^*(X; \mathbb{F}_p),; \mathbb{F}p) \implies \pi{t-s}(X)^\wedge_p$$
where $A$ is the mod-$p$ Steenrod algebra and the convergence is to the $p$-completed stable homotopy groups.
The $E_2$ page
htpy computes the $E_2$ page — that is, the Ext groups $\operatorname{Ext}_A^{s,t}(M, \mathbb{F}_p)$ for an $A$-module $M$ — by constructing a minimal free resolution of $M$ over $A$:
$$\cdots \to F_2 \xrightarrow{d_2} F_1 \xrightarrow{d_1} F_0 \xrightarrow{\varepsilon} M \to 0$$
where each $F_s$ is a free module over the Steenrod algebra. “Minimal” means the differentials have no linear (degree 0) component, ensuring the resolution is as small as possible. The dimension of the degree-$t$ part of $F_s$ equals $\dim \operatorname{Ext}_A^{s,t}(M, \mathbb{F}_p)$, so constructing the resolution directly gives the $E_2$ page.
Differentials
The $d_r$ differentials go
$$d_r : E_r^{s,t} \to E_r^{s+r, t+r-1}$$
In the $(t-s, s)$ chart, $d_r$ goes one step to the left and $r$ steps up. The first nontrivial differential $d_2$ can be computed algebraically via secondary Steenrod operations (following Bruner and Nassau). Higher differentials generally require geometric input or indirect arguments.
htpy’s deduction engine propagates known differentials using the Leibniz rule, naturality, Massey products (Moss’ theorem), and several other techniques. See Deduction Engine: Mathematical Justification for the complete list.
Products
The Adams spectral sequence carries a product structure induced by the Yoneda product in Ext:
$$\operatorname{Ext}_A^{s_1, t_1}(M, \mathbb{F}_p) \otimes \operatorname{Ext}_A^{s_2, t_2}(M, \mathbb{F}_p) \to \operatorname{Ext}_A^{s_1 + s_2,; t_1 + t_2}(M, \mathbb{F}_p)$$
htpy computes products with standard generators by composing chain maps along the resolution. At $p = 2$, the generators are $h_i \in \operatorname{Ext}^{1, 2^i}$ corresponding to the Hopf invariant one elements. The products are displayed as structlines in the viewer.
This product structure is essential for the Leibniz rule:
$$d_r(a \cdot x) = d_r(a) \cdot x + (-1)^{|a|} a \cdot d_r(x)$$
When $a$ is a permanent cycle ($d_r(a) = 0$ for all $r$), this simplifies to $d_r(a \cdot x) = \pm a \cdot d_r(x)$, allowing differentials to propagate multiplicatively through the spectral sequence.
Massey products
When $\alpha \beta = 0$ and $\beta \gamma = 0$ in Ext, the Massey product $\langle \alpha, \beta, \gamma \rangle$ is defined as a coset in Ext. htpy computes Massey products at the chain level using the Yoneda resolution and tracks indeterminacy explicitly. These are used by the deduction engine via Moss’ theorem to derive differentials from the $E_r$ Massey bracket structure.
The sphere spectrum at $p = 2$
The archive module S_2 contains the Adams spectral sequence for $S^0$ at $p = 2$:
- $E_2$ page: $\operatorname{Ext}_A(\mathbb{F}_2, \mathbb{F}_2)$ through stem 90
- $d_2$ differentials from the BFCC secondary resolution
- Higher differentials from the LWX machine proofs
- Products with $h_0, h_1, h_2, h_3, h_4$
- Named generators following classical conventions
The archive module S_3 contains the corresponding data at $p = 3$ through stem 120.