Spectral Sequences

Job: extract (co)homology of a target from a filtration or double complex by successive pages of approximations — the computational engine of homological algebra, topology, and geometry.

Core

A spectral sequence is a sequence of pages \((E_r,d_r)_{r\ge r_0}\) of bigraded modules with \(d_r^2=0\) and \(E_{r+1}\cong H(E_r,d_r)\), intended to converge to a graded object \(\mathrm{gr}\,H\) associated with a filtration on a limit \(H\).

Layer Content
Construction filtered complexes; exact couples; double complexes
Pages \(E_0\) (associated gr) \(\to E_1\) (homology) \(\to E_2\) …
Convergence regular filtrations; conditional / strong
Classics Serre, Leray, Grothendieck / hypercohomology, Lyndon–Hochschild–Serre

Reading map

Page Focus
Foundations filtered complexes, pages, convergence, Serre/Leray/Grothendieck, hypercohomology
Adams & Adams–Novikov Steenrod algebra Ext, stable stems, BP/chromatic
Eilenberg–Moore & Bar Tor/loop space, bar complex, Koszul duality
Motivic & Slice motivic Adams, Voevodsky slice, MGL, \(\mathbb{R}\)-motivic/\(C_2\)
Bockstein & Descent Bockstein mod-\(p\), Cartan–Leray, LHS, change-of-rings

Dependency order

  1. Homological algebra (complexes, Ext, derived functors).
  2. Algebraic topology for Serre.
  3. AG cohomology for Leray / hypercohomology.
  4. Worked examples until \(E_2\)-page reading is automatic.

Interfaces

Area Contact
Homological algebra Grothendieck SS for composed functors
Derived categories filtration \(\rightsquigarrow\) t-structures / Postnikov
Algebraic topology Serre, Atiyah–Hirzebruch, Adams
Algebraic geometry Leray, Hodge-de Rham, hypercohomology
K-theory Atiyah–Hirzebruch; motivic SS
Higher categories homotopy limits produce SS

Notation (paper-standard)

  • \(E_r^{p,q}\), \(d_r:E_r^{p,q}\to E_r^{p+r,q-r+1}\) (cohomological convention)
  • \(E_2^{p,q}\Rightarrow H^{p+q}\) convergence
  • \({}'E\), \({}''E\) for first/second filtration on a double complex

See it

Universal coefficient and Künneth as collapsed SS. The Künneth spectral sequence \(\mathrm{Tor}_p(H_q(C),H_r(D))\Rightarrow H_{p+q+r}(C\otimes D)\) (under flatness hyp.\ collapses to the short exact Künneth sequence). Training ground before Leray.

Check

What does \(E_2^{p,q}\Rightarrow H^{n}\) mean operationally for computing \(H^n\)?

There is a filtration \(F^\bullet H^n\) with \(\mathrm{gr}^p H^n\cong E_\infty^{p,n-p}\); each \(E_\infty\) is a subquotient of \(E_2\) obtained by killing successive \(d_r\)-cycles/boundaries. Without knowing differentials, \(E_2\) only bounds / constrains \(H\).

Why can a spectral sequence fail to determine \(H\) even if all \(d_r=0\) for \(r\ge 2\)?

Extension problems: \(E_\infty^{p,n-p}\cong\mathrm{gr}^p H^n\) need not split uniquely as \(H^n\cong\bigoplus_p E_\infty^{p,n-p}\); nontrivial extensions among filtration layers remain.

Pitfalls

  • Cohomological vs homological indexing flip differential bidegrees — always check the paper’s convention.
  • Conditional convergence (Boardman) in unbounded filtrations: naive \(\lim E_r\) may mislead.
  • “Degenerates at \(E_2\)” means \(d_r=0\) for \(r\ge 2\), not that \(E_2=H\).

Refs

  • Weibel, An Introduction to Homological Algebra (Ch.\ 5)
  • McCleary, A User’s Guide to Spectral Sequences
  • Boardman, Conditionally convergent spectral sequences
  • Stacks Project, Spectral sequences

Next

Foundations · Homological algebra · Derived categories · Algebraic topology · AG cohomology · Adams & Adams–Novikov · Eilenberg–Moore & Bar · Motivic & Slice · Bockstein & Descent