+++ title = “The Cartan-Eilenberg Spectral Sequence” section = “mathematics” +++

The Cartan–Eilenberg Spectral Sequence

The Cartan–Eilenberg spectral sequence (CESS) arises from the factorization of the Steenrod algebra into its polynomial and exterior parts. It provides a way to compute the Adams $E_2$ page from simpler pieces.

Setup

The dual Steenrod algebra $A_*$ at $p = 2$ factors as

$$A_* \cong P_* \otimes E_*$$

where $P_* = \mathbb{F}2[\xi_1, \xi_2, \ldots]$ is the polynomial part and $E* = E(\tau_0, \tau_1, \ldots)$ is the exterior part. Dually, the Steenrod algebra $A$ has subalgebras $P^$ and quotient $E^ = A // P^*$.

The CESS takes the form

$$E_2^{s,u} = \operatorname{Ext}_{P^}^{s}(\mathbb{F}2,; \operatorname{Ext}{E^}^{u}(\mathbb{F}_2, \mathbb{F}_2)) \implies \operatorname{Ext}_A^{s+u}(\mathbb{F}_2, \mathbb{F}_2)$$

where the target is the Adams $E_2$ page. Since $\operatorname{Ext}_{E^*}(\mathbb{F}_2, \mathbb{F}_2) \cong \mathbb{F}_2[q_0, q_1, q_2, \ldots]$ (a polynomial algebra on classes $q_i$ of degree $2^{i+1} - 1$), the CESS $E_2$ page is

$$\operatorname{Ext}_{P^*}(\mathbb{F}_2,; \mathbb{F}_2[q_0, q_1, \ldots])$$

What htpy computes

htpy resolves $\mathbb{F}_2[q_0, \ldots, q_N]$ as a module over $P^*$ using a minimal free resolution. This gives the CESS $E_2$ page directly. The resolution carries a CE-filtration (the $u$-grading from the exterior part) and a weight grading from the monomial structure of the polynomial module.

The same $E_2$ page serves double duty:

• As the CESS $E_2$ page converging to $\operatorname{Ext}_A(\mathbb{F}_2, \mathbb{F}_2)$ (the Adams $E_2$ page)
• As the algebraic Novikov spectral sequence (aNSS) $E_2$ page converging to $\operatorname{Ext}{BPBP}(BP_, BP_*)$ (the Adams–Novikov $E_2$ page)

The two spectral sequences share the same $E_2$ page but have different differentials.

CESS differentials

The CESS differentials $d_r : E_r^{s,u} \to E_r^{s+r, u-r+1}$ arise from the extension structure $P^* \to A \to E^*$ of Hopf algebras. In Adams coordinates, $d_r$ shifts by $(-1, +r)$ in the $(n, s)$ chart.

htpy computes CESS differentials via the deduction engine, using the degree argument, Leibniz propagation, convergence constraints (CESS $E_\infty$ must equal Adams $E_2$), and elimination by contradiction.

Cross-validation

The CESS computation is validated by two key checks:

• V1b: Per-stem totals of CESS $E_2$ dimensions are $\geq$ Adams $E_2$ dimensions (the CESS can only lose classes through differentials).
• V1c: CESS weight counts at each bidegree match the Ext dimension (weight decomposition is consistent).

The archive module S_2_cess

The archive module S_2_cess contains:

• The CESS/aNSS $E_2$ page through stem 30, filtration 16
• Computed using $q_0$ through $q_5$
• Generator names matched from the algebraic Novikov CSV where possible
• Weight annotations per generator

Limitations

• Currently $p = 2$ only. Odd-prime CESS requires generalizing the $P^*$ algebra degree formulas and the bitmask basis encoding.
• Algebraic CESS $d_2$ (from the Hopf extension structure, Ravenel A1.3.16) is not yet computed directly; instead, CESS differentials are derived via the deduction engine.