Spectral Sequences: Foundations¶

Job: construct spectral sequences from filtered complexes and double complexes; state convergence; package Serre, Leray, Grothendieck, and hypercohomology spectral sequences.

Core¶

Filtered complexes¶

Let $C^\bullet$ be a cochain complex with a decreasing filtration $F^p C^\bullet$ compatible with $d$ ($dF^p\subset F^p$). The associated graded is $$ \mathrm{gr}^p C^n=F^p C^n/F^{p+1}C^n. $$ Assumptions for good behavior: filtration exhaustive ($\bigcup_p F^p=C$ or $\bigcap_p F^p=0$ depending on increasing/decreasing conventions) and complete / Hausdorff in the filtration topology when unbounded.

Spectral sequence of a filtered complex (cohomological). $$ E_0^{p,q}=\mathrm{gr}^p C^{p+q},\qquad d_0=\mathrm{gr}\,d, $$ $$ E_1^{p,q}=H^{p+q}(\mathrm{gr}^p C^\bullet),\qquad E_2^{p,q}=H^p_F\bigl(H^{q}(\mathrm{gr}\,C)\bigr)\ \text{(schematic)}, $$ more precisely $E_1^{p,q}=H^{p+q}(F^p/F^{p+1})$ with $d_1$ induced by $d$, and $$ E_r^{p,q}\Rightarrow H^{p+q}(C^\bullet) $$ under regularity (for each $n$, only finitely many $p$ with $E_r^{p,n-p}\neq 0$ for large $r$, or Boardman conditions).

Page	Meaning
$E_0$	graded pieces of the complex
$E_1$	cohomology of graded pieces
$E_2$	often “cohomology of cohomology”
$E_\infty$	$\mathrm{gr}^p H^{p+q}(C)$

Exact couples¶

An exact couple $(D,E;i,j,k)$ with exact triangle $D\xrightarrow{i}D\xrightarrow{j}E\xrightarrow{k}D$ produces $d=jk$ on $E$ and a derived couple; iterating yields a spectral sequence. Filtered complexes give exact couples via $D^{p,q}=H^{p+q}(F^p C)$, $E^{p,q}=H^{p+q}(F^p/F^{p+1})$.

Double complexes¶

A double complex $C^{p,q}$ with $d_h,d_v$, $d_h d_v+d_v d_h=0$, total complex $\mathrm{Tot}^\oplus$ or $\mathrm{Tot}^\prod$. Two filtrations:

\[ {}'F^p\mathrm{Tot}^n=\bigoplus_{i\ge p}C^{i,n-i},\qquad {}''F^q\mathrm{Tot}^n=\bigoplus_{j\ge q}C^{n-j,j}. \]

First quadrant double complex: both spectral sequences converge to $H(\mathrm{Tot})$: $$ {}'E_2^{p,q}=H^p_h H^q_v(C)\Rightarrow H^{p+q}(\mathrm{Tot}),\qquad {}''E_2^{p,q}=H^q_v H^p_h(C)\Rightarrow H^{p+q}(\mathrm{Tot}). $$ Standard machine for balancing $\mathrm{Tor}$ / $\mathrm{Ext}$ and for hypercohomology.

Convergence¶

Regular filtration (for each $n$, $F^p H^n=0$ for $p\gg 0$ and $F^p H^n=H^n$ for $p\ll 0$ in cohomological decreasing conventions) $\Rightarrow$ strong convergence $E_\infty^{p,q}\cong\mathrm{gr}^p H^{p+q}$.

Conditional convergence (Boardman): more delicate $\lim^1$ issues when filtrations are infinite in both directions. Unbounded derived categories need care (Spaltenstein resolutions, etc.).

Extension problems. Even with known $E_\infty$, reconstructing $H^n$ from $\mathrm{gr}\,H^n$ may involve nontrivial extensions in the abelian category of coefficients.

Serre spectral sequence¶

For a Serre fibration $F\to E\to B$ (path-connected base, typical orientation hyp.), $$ E_2^{p,q}=H^p\bigl(B;\mathcal{H}^q(F;G)\bigr)\Rightarrow H^{p+q}(E;G) $$ (local coefficients $\mathcal{H}^q(F)$ if $\pi_1(B)$ acts). Homological version with $H_p(B;H_q(F))$.

Uses. Compute $H^*(SU(n))$, loop spaces; relate $H^*(BG)$ and $H^*(G)$; Gysin sequences when fibers are spheres.

Leray spectral sequence¶

For a continuous map $f:X\to Y$ and sheaf $\mathcal{F}$ on $X$, $$ E_2^{p,q}=H^p\bigl(Y;R^q f_*\mathcal{F}\bigr)\Rightarrow H^{p+q}(X;\mathcal{F}). $$ If $f$ is a fibration with fiber $F$ and local system conditions, recovers Serre. In algebraic geometry, $f:X\to Y$ morphism of schemes, $\mathcal{F}$ étale/sheaf of modules: same shape for étale or Zariski cohomology under hypotheses — see AG cohomology.

Grothendieck spectral sequence¶

For functors $G:\mathcal{A}\to\mathcal{B}$, $F:\mathcal{B}\to\mathcal{C}$ between abelian categories with enough injectives, if $G$ sends injectives to $F$-acyclic objects, $$ E_2^{p,q}=(R^p F)(R^q G)(A)\Rightarrow R^{p+q}(FG)(A). $$

Instance	$F$, $G$
Leray	$\Gamma(Y,-)$, $f_*$
Lyndon–Hochschild–Serre	$H^(Q,-)$, $H^(N,-)$ for $1\to N\to G\to Q\to 1$
Change of rings	$\mathrm{Ext}_S$, restriction
Composition of derived pushforwards	$g_\circ f_$

Hypercohomology¶

For a complex of sheaves $\mathcal{F}^\bullet$, hypercohomology $\mathbb{H}^n(X,\mathcal{F}^\bullet)=H^n(R\Gamma(X,\mathcal{F}^\bullet))$. Two spectral sequences from the double complex of an injective Cartan–Eilenberg resolution: $$ {}'E_1^{p,q}=H^q(X;\mathcal{F}^p)\Rightarrow\mathbb{H}^{p+q}(X,\mathcal{F}^\bullet), $$ $$ {}''E_2^{p,q}=H^p\bigl(X;\mathcal{H}^q(\mathcal{F}^\bullet)\bigr)\Rightarrow\mathbb{H}^{p+q}(X,\mathcal{F}^\bullet). $$ Hodge-to-de Rham. $\mathcal{F}^\bullet=\Omega^\bullet_{X/\mathbb{C}}$: $E_1^{p,q}=H^q(X,\Omega^p)\Rightarrow H^{p+q}_{\mathrm{dR}}(X)$ (degenerates at $E_1$ for smooth proper $X/\mathbb{C}$ in char $0$ — Deligne–Illusie in char $p$ lifting).

See it¶

Computing $H^*(\mathbb{CP}^\infty)$ via Serre. Path fibration $\Omega BS^1\to PS^1\to BS^1$ with $BS^1\simeq\mathbb{CP}^\infty$, fiber $S^1$: Serre SS yields polynomial algebra $\mathbb{Z}[c]$ on the first Chern class — model for characteristic class computations.

Grothendieck SS for $\Gamma\circ f_*$. $R\Gamma(X,-)=R\Gamma(Y,Rf_*-)$ gives Leray as the special case $E_2^{p,q}=H^p(Y,R^qf_*\mathcal{F})\Rightarrow H^{p+q}(X,\mathcal{F})$. Entire modern AG cohomology calculus is organized this way.

Check¶

Set up the double complex spectral sequences that prove $\mathrm{Tor}$ is balanced.

Take projective resolutions $P_\bullet\to M$, $Q_\bullet\to N$; form double complex $P_p\otimes Q_q$. ${}'E_2$ computes $H_*(P\otimes N)\cong\mathrm{Tor}_*(M,N)$ via resolving $M$; ${}''E_2$ via $N$. Both abut to $H(\mathrm{Tot})$, hence equal.

In the Serre SS for a trivial product fibration $B\times F\to B$, why does it collapse under field coefficients with finite-dim homology?

Local system trivial; $E_2=H^*(B)\otimes H^*(F)$ (field coeffs). For products, the SS is multiplicative and agrees with the Künneth isomorphism already at $E_2$ with $d_r=0$ when Künneth applies — differentials vanish because the projection and inclusion split the cohomology ring structure. (If not a product, differentials encode the twisting.)

Pitfalls¶

Filtration degree vs cohomological degree: $p$ is filtration; total degree $n=p+q$.
$\mathrm{Tot}^\oplus$ vs $\mathrm{Tot}^\prod$: matter for unbounded double complexes; choose the one that computes the correct derived functor.
Local coefficients in Serre are mandatory when $\pi_1(B)$ acts nontrivially on $H^*(F)$ — forgetting them is a classic error.
Degeneration $\neq$ collapse of extension problems at $E_\infty$.

Refs¶

Weibel, Homological Algebra
McCleary, A User’s Guide to Spectral Sequences
Bott–Tu, Differential Forms in Algebraic Topology (Serre SS geometry)
Godement, Topologie algébrique et théorie des faisceaux (hypercohomology)

Next¶

Spectral sequences index · Homological algebra · Derived categories · Algebraic topology · AG cohomology · K-theory · Adams & Adams–Novikov · Eilenberg–Moore & Bar · Motivic & Slice · Bockstein & Descent