Spectral Sequence

Overview

A spectral sequence packages a succession of cohomology computations. Its early pages usually involve simpler pieces of a problem, while later differentials record interactions between those pieces. When a convergence theorem applies, the surviving terms describe a filtration on the desired answer. This article uses the cohomological convention and focuses on spectral sequences arising from filtered complexes of modules.

Prerequisites

Understand chain complexes, their cochain reindexing, and homology as kernel modulo image. A decreasing filtration is a family of submodules $\cdots\supseteq F^pC\supseteq F^{p+1}C\supseteq\cdots$. Its associated graded pieces are $\operatorname{gr}^pC=F^pC/F^{p+1}C$. No derived-category formalism is needed for the construction described here.

Definition and conventions

A cohomological bigraded spectral sequence of left modules over a fixed unital ring $R$ consists of modules $E_r^{p,q}$, for integers $p,q$ and pages $r\geq r_0$, with square-zero differentials

and specified isomorphisms

The total degree $p+q$ increases by one under $d_r$. Abstract spectral sequences can instead be defined using differential objects without bigradings; convergence to a target is additional information.

For a cochain complex $C^\bullet$ with $d(F^pC^n)\subseteq F^pC^{n+1}$, its filtration gives a spectral sequence starting with

Different sources start numbering at different pages or use homological indices; the differential convention must be checked before comparing formulas.

Intuition

The first approximation sees only the part of the differential that stays at the same filtration level. A class surviving that approximation may still fail to be a genuine cocycle: its differential can lie deeper in the filtration. Later pages detect these increasingly distant failures. Taking cohomology discards classes killed by outgoing differentials and identifies those hit by incoming differentials.

Examples and non-examples

Let $k$ be a field and let $C^0=ka$, $C^1=kb$, with $d(a)=b$ and all other terms zero. Give $a$ filtration degree zero and $b$ filtration degree two: $F^pC^0=ka$ for $p\leq0$, zero otherwise, and $F^pC^1=kb$ for $p\leq2$, zero otherwise. This filtration is preserved by $d$.

The only nonzero $E_0$ terms occur at $(0,0)$ and $(2,-1)$. Both $d_0$ and $d_1$ vanish because their target bidegrees contain no possible nonzero terms. Yet

is an isomorphism, induced by $d(a)=b$. Hence $E_3=0$, agreeing with the zero cohomology of $C$. Vanishing of one page's differentials therefore does not imply vanishing of all subsequent differentials.

A sequence of bigraded modules with arbitrarily selected square-zero maps is not automatically a spectral sequence: the next page must be identified with the preceding page's cohomology. A collection of ranks without those maps and identifications is insufficient.

Important properties

Suppose the filtration on each $C^n$ is finite: for each $n$ there are integers $a(n),b(n)$ with $F^{a(n)}C^n=C^n$ and $F^{b(n)}C^n=0$. Then the filtered-complex spectral sequence converges, with

where $F^pH^n$ is the image of $H^n(F^pC)\to H^n(C)$. Proof idea: finiteness ensures that the successive conditions of being a cycle modulo deeper filtration eventually become actual cocycle conditions, and all relevant boundaries eventually appear. Unbounded filtrations require separate convergence arguments.

Even after convergence, graded pieces need not determine the target module. The abelian groups $\mathbb Z/4$ and $(\mathbb Z/2)^2$ both admit filtrations with two successive quotients $\mathbb Z/2$, although the groups are not isomorphic. Such extension information survives beyond the page calculation.

Connections and further reading

Double complexes and compositions of derived functors produce important spectral sequences used in Ext and sheaf cohomology. Read Li's general definitions alongside Stacks, Section 12.24, keeping the finite-filtration hypothesis in view.