+++ title = “Spectral Sequences” section = “mathematics” +++
Spectral Sequences
A spectral sequence is an algebraic gadget that computes homological invariants through a sequence of successive approximations. It consists of a sequence of pages $(E_r, d_r)$ for $r = 2, 3, 4, \ldots$, where each page is a bigraded abelian group and $d_r$ is a differential ($d_r \circ d_r = 0$) whose homology gives the next page:
$$E_{r+1}^{s,t} = \frac{\ker(d_r : E_r^{s,t} \to E_r^{s+r, t+r-1})}{\operatorname{im}(d_r : E_r^{s-r, t-r+1} \to E_r^{s,t})}$$
The idea is that $E_2$ is something computable (typically an Ext group), the differentials $d_r$ encode obstructions, and the spectral sequence converges to the object you actually want to compute.
Why spectral sequences?
Many problems in algebraic topology reduce to extension problems that are too hard to solve directly. Spectral sequences break these problems into a sequence of manageable steps. The trade-off is bookkeeping: tracking which classes survive, which differentials are nonzero, and how the pages relate to each other.
This is where computational tools like htpy become valuable. The combinatorial data in a spectral sequence — dimensions, differentials, product structures — is well-suited to machine computation and visualization.
The spectral sequences htpy computes
htpy works with several interrelated spectral sequences:
| Spectral sequence | $E_2$ page | Converges to | Status |
|---|---|---|---|
| Classical Adams | $\operatorname{Ext}_A^{s,t}(\mathbb{F}_p, \mathbb{F}_p)$ | $\pi_{t-s}(S^0)^\wedge_p$ | Full computation |
| Motivic Adams | $\operatorname{Ext}{A{\mathrm{mot}}}^{s,t,w}(\mathbb{F}_2, \mathbb{F}_2)$ | $\pi_{t-s,w}(S^{0,0}_{\mathrm{mot}})^\wedge_2$ | Full computation |
| Algebraic Novikov | $\operatorname{Ext}{P*}(\mathbb{F}_2, \mathbb{F}_2[q_0, \ldots])$ | $\operatorname{Ext}{BPBP}(BP_, BP_*)$ | $E_2$ computed, diffs from CSV |
| CESS | $\operatorname{Ext}{P*}(\mathbb{F}_2, \mathbb{F}_2[q_0, \ldots])$ | $\operatorname{Ext}_A(\mathbb{F}_2, \mathbb{F}_2)$ | $E_2$ computed |
| $C\tau$ Adams | $\operatorname{Ext}{A{\mathrm{mot}}}(C\tau, \mathbb{F}_2)$ | (collapses at $E_2$) | Full computation |
These are connected by natural maps, comparison theorems, and the Gheorghe–Wang–Xu isomorphism $\operatorname{Ext}{A{\mathrm{mot}}}(C\tau, \mathbb{F}2) \cong \operatorname{Ext}{BP_BP}(BP_, BP_*)$. See the Motivic Adams and $C\tau$ Bridge sections for details.
Chart conventions
htpy displays spectral sequences in the $(t-s, s)$ plane:
- The stem $n = t - s$ is the horizontal axis
- The Adams filtration $s$ is the vertical axis
Differentials $d_r : E_r^{s,t} \to E_r^{s+r, t+r-1}$ go one step to the left and $r$ steps up in this chart. Product structlines connect classes related by multiplication with standard generators ($h_0, h_1, h_2, \ldots$ at $p = 2$).