nLab
spectral sequence

under construction

Idea

A cohomology spectral sequence is a sequence of certain cochain complexes such that each is in each entry the cochain cohomology of the previous one.

Spectral sequences are used as computational tools in homological algebra and more generally in homotopy theory. Notably they are used to compute composition of derived functors.

Traditionally this is considered on model categories of chain complexes in some abelian category for which fibrant replacement is given by injective resolution of chain complexes. But more generally there is a notion of nonabelian/unstable spectral sequences, called homotopy spectral sequences.

Definition

Throughout, let $𝒜$ be an abelian category.

Spectral sequence

Definition

A cohomology spectral sequence in $𝒜$ is

• a family $(E_r^{p,q})$ of objects in $𝒜$, for all integers $p,q,r$ with $r\ge 1$

  (for a fixed $r$ these are said to form the $r$-th page of the spectral sequence)

• for each $p,q,r$ as above a morphism (called the differential)

  $$d_r^{p,q}:E_r^{p,q}\to E_r^{p+r,q-r+1}$$d^{p,q}_r:E^{p,q}_r\to E^{p+r,q-r+1}_r

  satisfying $d_r^2=0$ (more precisely, $d_r^{p+r,q-r+1}\circ d_r^{p,q}=0$

• isomorphisms ${\alpha }_r^{p,q}:H^{p,q}(E_r)\to E_{r+1}^{p,q}$ where the chain cohomology is given by

  $$H^{p,q}(E_r)=\ker d_r^{p,q}/\mathrm{im}{d}_r^{p-r,q+r-1}.$$H^{p,q}(E_r) = \mathrm{ker} d^{p,q}_r/ \mathrm{im} d^{p-r,q+r-1}_r \,.

Analogously a homology spectral sequence is collection of objects $(E_{p,q}^r)$ with the differential $d_r$ of degree $(-r,r-1)$.

Convergence

Definition

(convergence)

A component of a spectral sequence $(E_r)$ converges at $(p,q)$ if there exists finite $r$ such that $E_r^{p,q}\simeq E_{r+1}^{p,q}\cdots =:E_{\mathrm{\infty }}^{p,q}$. One writes this as

$$E_r^{p,q}\Rightarrow E_{\mathrm{\infty }}^{p,q}.$$E^{p,q}_r \Rightarrow E^{p,q}_\infty \,.

A spectral sequence is said to degenerate in the $E_r$-term if $d_{r\prime }^{p,q}=0$ for all $r\prime \ge r$. Then clearly it converges degreewise.

A spectral sequence is called bounded if for each $r$ and $n$ there are only finitely many non-vanishing terms of the form $E_r^{p,n-p}$. Also a bounded spectral sequence converges degreewise.

A spectral sequence $(E_r)$ converges if it converges degreewise to a graded filtration: if there is a graded object $(H^n{)}_{n\in \mathbb{Z}}$ equipped in each degree with a finite filtration

$$0\subset \cdots \subset F^p{H}^n\subset F^{p-1}{H}^n\subset \cdots \subset F^0{H}^n:=H^n$$0 \subset \cdots \subset F^{p} H^n \subset F^{p-1}H^n \subset \cdots \subset F^0 H^n := H^n

such that

$$E_{\mathrm{\infty }}^{p,q}\simeq F^p{H}^{p+q}/F^{p+1}{H}^{p+q}.$$E_\infty^{p,q} \simeq F^p H^{p+q} / F^{p+1}H^{p+q} \,.

The notation for this is

$$E_r^{p,q}\Rightarrow H^{p+q}.$$E_r^{p,q} \Rightarrow H^{p+q} \,.

Remark

In applications one is interested in computing the $H^n$ and uses spectral sequences converging to this as tools for approximating $H^n$ in terms of the given filtration.

Therefore usually spectral sequences are required to converge in each degree, or even that for each pair $(p,q)$ there exists an $r_0$ such that for all $r\ge r_0$, $d_r^{p-r,q+r-1}=0$.

Definition

(Collapse)

A spectral sequence collapses at $r$ if in $E_r^{p,q}$ only a single row or a single column in non-vanishing.

Observation

If $(E_r)$ collapses at $r$, then it converges to $H^{•}$ with $H^n$ being the unique entry $E_r^{p,q}$ on the non-vanishing row/column with $p+q=n$.

Graphical presentation

(…)

Examples

First quadrant spectral sequence

Definition

A first quadrant spectral sequence is one for wich all pages are concentrated in the first quadrant of the $(p,q)$-plane, in that

$$((p<0)\mathrm{or}(q<0))\Rightarrow E_r^{p,q}=0.$$((p \lt 0) or (q \lt 0)) \;\; \Rightarrow E_r^{p,q} = 0 \,.

Observation

If the $r$th page is concentrated in the first quadrant, then so the $(r+1)\mathrm{st}$ page. So if the first one is, then all are.

Observation

Every first quadrant spectral sequence converges at $(p,q)$ from $r>\max (p,q+1)$ on

$$E_{\max (p,q+1)+1}^{p,q}=E_{\mathrm{\infty }}^{p,q}.$$E_{max(p,q+1)+1}^{p,q} = E_\infty^{p,q} \,.

Observation

If a first quadrant spectral sequence converges

$$E_r^{p,q}\Rightarrow H^{p+q}$$E_r^{p,q} \Rightarrow H^{p+q}

then each $H^n$ has a filtration of length $n+1$

$$0=F^{n+1}{H}^n\subset F^n{H}^n\subset \cdots \subset F^1{H}^n\subset F^0{H}^n=H^n$$0 = F^{n+1}H^n \subset F^n H^n \subset \cdots \subset F^1 H^n \subset F^0 H^n = H^n

and we have

• $F^n{H}^n\simeq E_{\mathrm{\infty }}^{n,0}$

• $H^n/F^1{H}^n\simeq E_{\mathrm{\infty }}^{0,n}$.

Spectral sequence of a filtered complex

If a cochain complex $C^{•}$ is given a filtration $F^{\star }{C}^{•}$, there is an induced filtration $F^{*}H(C)$ of its cohomology, according to which levels of the filtration contain representatives for the various cohomology classes.

A filtration $F$ also gives rise to a graded object $\mathrm{Gr}(F)$, whose grades are the successive level inclusion cokernels. Generically, the operations of grading and homology do not commute: $\mathrm{Gr}(F^{*}{H}^{•}(C))\ne H^{•}(\mathrm{Gr}(F^{\star })C)$.

As explained more fully at this other page, there is a spectral sequence associated to a filtered complex $F^{\star }{C}^{•}$, passing through $H^{•}(\mathrm{Gr}(F^{\star })C)$ in the page $E_{(1)}$ and which in good cases converges to $\mathrm{Gr}(F^{*}{H}^{•}(C))$.

Spectral sequence of a double complex

A double complex is naturally filtered in two ways: by columns and by rows. By the above this gives two different spectral sequences associated with it.

(…)

Exact couples

An exact couple is an exact sequence of three arrows among two objects

$$E\stackrel{j}{\to }D\stackrel{\phi }{\to }D\stackrel{k}{\to }E\stackrel{j}{\to }.$$E \overset{j}{\to} D \overset{\varphi}{\to} D \overset{k}{\to} E \overset{j}{\to}.

These creatures construct spectral sequences by a two-step process:

• first, the composite $d=\mathrm{kj}:E\to E$ is nilpotent: $d^2=0$

• second, the homology $E\prime$ of $(E,d)$ supports a map $j\prime :E\prime \to \phi D$, and receives a map $k\prime :\phi D\to E\prime$. Setting $D\prime =\phi D$, by general nonsense

  $$E\prime \stackrel{j\prime }{\to }D\prime \stackrel{\phi }{\to }D\prime \stackrel{k\prime }{\to }E\prime \stackrel{j\prime }{\to }.$$E' \overset{j'}{\to} D' \overset{\varphi}{\to} D' \overset{k'}{\to} E' \overset{j'}{\to}.

  is again an exact couple.

The sequence of complexes $(E,d),(E\prime ,d\prime ),\dots$ is a spectral sequence, by construction.

Examples of exact couples can be constructed in a number of ways. Importantly, any short exact sequence involving two distinct chain complexes provides an exact couple among their total homology complexes, via the Mayer-Vietoris long exact sequence; in particular, applying this procedure to the relative homology of a filtered complex gives precisely the spectral sequence of the filtered complex described (???) somewhere else on this page. For another example, choosing a chain complex of flat modules $(C^{\dot{,}}d)$, tensoring with the short exact sequence

$$\mathbb{Z}/p\mathbb{Z}\to \mathbb{Z}/p^2\mathbb{Z}\to \mathbb{Z}/p\mathbb{Z}$$\mathbb{Z}/p\mathbb{Z} \to \mathbb{Z}/p^2\mathbb{Z} \to \mathbb{Z}/p\mathbb{Z}

gives the exact couple

$$H^{\dot{(}}d,\mathbb{Z}/p^2\mathbb{Z})\stackrel{[\cdot ]}{\to }{H}^{\dot{(}}d,\mathbb{Z}/p\mathbb{Z})\stackrel{\beta }{\to }{H}^{\dot{(}}d,\mathbb{Z}/p\mathbb{Z})\stackrel{p}{\to }{H}^{\dot{(}}d,\mathbb{Z}/p^2\mathbb{Z})\cdots$$H^\dot(d,\mathbb{Z}/p^2\mathbb{Z})\overset{[\cdot]}{\to} H^\dot(d,\mathbb{Z}/p\mathbb{Z})\overset{\beta}{\to}H^\dot(d,\mathbb{Z}/p\mathbb{Z})\overset{p}{\to}H^\dot(d,\mathbb{Z}/p^2\mathbb{Z})\cdots

in which $\beta$ is the mod-$p$ Bockstein homomorphism.

The exact couple recipe for spectral sequences is notable in that it doesn’t mention any grading on the objects $D,E$; trivially, an exact couple can be specified by a short exact sequence $coker\phi \to E\to \ker \phi$, although this obscures the focus usually given to $E$. In applications, a bi-grading is usually induced by the context, which also specifies bidegrees for the initial maps $j,k,\phi$, leading to the conventions mentioned earlier.

Spectral sequences for hyper-derived functors

(…)

Grothendieck spectral sequence

Let $𝒜\stackrel{F}{\to }ℬ\stackrel{G}{\to }𝒞$ be two left exact functors between abelian categories.

Write $R^{p}F:𝒟\to \mathrm{Ab}$ for the cochain cohomology of the derived functor of $F$ in degree $p$ etc.

Theorem

If $F$ sends injective objects of $𝒜$ to $G$-acyclic objects in $ℬ$ then for each $A\in 𝒜$ there is a first quadrant cohomology spectral sequence

$$E_r^{p,q}:=(R^{p}G\circ R^{q}F)(A)$$E_r^{p,q} := (R^p G \circ R^q F)(A)

that converges to the right derived functor of the composite functor

$$E_r^{p,q}\Rightarrow R^{p+q}(G\circ F)(A).$$E_r^{p,q} \Rightarrow R^{p+q} (G \circ F)(A).

Moreover

1. the edge maps in this spectral sequence are the canonical morphisms

   $$R^{p}G(FA)\to R^p(G\circ F)(A)$$R^p G (F A) \to R^p (G \circ F)(A)

   induced from applying $F$ to an injective resolution $A\to \hat{A}$ and the isomorphism

   $$R^q(G\circ F)(A)\to G(R^{q}F(A)).$$R^q (G \circ F)(A) \to G(R^q F (A)) \,.

2. the exact sequence of low degree terms is

   $$0\to (R^{1}G)(F(A))\to R^1(G\circ F)(A)\to F(R^1(G(A)))\to (R^{2}F)(G(A))\to R^2(G\circ F)(A)$$0 \to (R^1 G)(F(A)) \to R^1(G \circ F)(A) \to F(R^1(G(A))) \to (R^2 F)(G(A)) \to R^2(G \circ F)(A)

Proof

Since for $A\to \hat{A}$ an injective resolution of $A$ the complex $F(\hat{A})$ is a chain complex not concentrated in a single degree, we have that $R^p(G\circ F)(A)$ is equivalently the hyper-derived functor evaluation ${ℝ}^p(G)(F(A))$.

Therefore the second spectral sequence discussed at hyper-derived functor spectral sequences converges as

$$(R^{p}G)H^q(F(\hat{A}))\Rightarrow R^p(G\circ F)(A).$$(R^p G)H^q(F(\hat A)) \Rightarrow R^p (G \circ F)(A) \,.

Now since by construction $H^q(F(\hat{A}))=R^{q}F(A)$ this is a spectral sequence

$$(R^{p}G)(R^{q}F)A)\Rightarrow R^p(G\circ F)(A).$$(R^p G)(R^q F) A) \Rightarrow R^p (G \circ F)(A) \,.

This is the Grothendieck spectral sequence.

Leray spectral sequence

The Leray spectral sequence is the special case of the Grothendieck spectral sequence for the case where the two functors being composed are a push-foward of sheaves of a abelian groups along a continuous map $f:X\to Y$ followed by the push-forward $X\to *$ to the point. This yields a spectral sequence that computes the abelian sheaf cohomology on $X$ in terms of the abelian sheaf cohomology on $Y$.

Theorem

Let $X,Y$ be suitable sites and $f:X\to Y$ be a morphism of sites. () Let $𝒞={\mathrm{Ch}}_{•}(\mathrm{Sh}(X,\mathrm{Ab}))$ and $𝒟={\mathrm{Ch}}_{bulle}(\mathrm{Sh}(Y,\mathrm{Ab}))$ be the model categories of complexes of sheaves of abelian groups. The direct image $f_{*}$ and global section functor ${\Gamma }_Y$ compose to ${\Gamma }_X$:

$${\Gamma }_X:𝒞\stackrel{f_{*}}{\to }𝒟\stackrel{{\Gamma }_Y}{\to }{\mathrm{Ch}}_{•}(\mathrm{Ab}).$$\Gamma_X : \mathcal{C} \stackrel{f_*}{\to} \mathcal{D} \stackrel{\Gamma_Y}{\to} Ch_\bullet(Ab) \,.

Then for $A\in \mathrm{Sh}(X,\mathrm{Ab})$ a sheaf of abelian groups on $X$ there is a cohomology spectral sequence

$$E_r^{p,q}:=H^p(Y,R^q{f}_{*}A)$$E_r^{p,q} := H^p(Y, R^q f_* A)

that converges as

$$E_r^{p,q}\Rightarrow H^{p+q}(X,A)$$E_r^{p,q} \Rightarrow H^{p+q}(X, A)

and hence computes the cohomology of $X$ with coefficients in $A$ in terms of the cohomology of $Y$ with coefficients in the push-forward of $A$.

Base change spectral sequence for $\mathrm{Tor}$ and $\mathrm{Ext}$

For $R$ a ring write $R$Mod for its category of modules. Given a homomorphism of rings $f:R_1\to R_2$ and an $R_2$-module $N$ there are composites of base change along $f$ with the hom-functor and the tensor product functor

$$R_1\mathrm{Mod}\stackrel{{\otimes }_{R_1}{R}_2}{\to }{R}_2\mathrm{Mod}\stackrel{{\otimes }_{R_2}N}{\to }\mathrm{Ab}$$R_1 Mod \stackrel{\otimes_{R_1} R_2}{\to} R_2 Mod \stackrel{\otimes_{R_2} N}{\to} Ab

$$R_1\mathrm{Mod}\stackrel{{\mathrm{Hom}}_{R_1\mathrm{Mod}}(-,R_2)}{\to }{R}_2\mathrm{Mod}\stackrel{{\mathrm{Hom}}_{R_2}(-,N)}{\to }\mathrm{Ab}.$$R_1 Mod \stackrel{Hom_{R_1 Mod}(-,R_2)}{\to} R_2 Mod \stackrel{Hom_{R_2}(-,N)}{\to} Ab \,.

The derived functors of ${\mathrm{Hom}}_{R_2}(-,N)$ and ${\otimes }_{R_2}N$ are the Ext- and the Tor-functors, respectively, so the Grothendieck spectral sequence applied to these composites yields a base change spectral sequence for these.

Properties

Lemma

(mapping lemma)

If $f:(E_r^{p,q}\to (F_r^{p,q}))$ is a morphism of spectral sequences such that for some $r$ we have that $f_r:E_r^{p,q}{toF}_r^{p,q}$ is an isomorphism, then also $f_s$ is an isomorphism for all $s\ge r$.

Lemma

(classical convergence theorem)

(…)

This is recalled in (Weibel, theorem 5.51).

References

Abelian/stable theory

A standard textbook reference is chapter 5 of

• Charles Weibel, An introduction to homological algebra Cambridge studies in advanced mathematics 38 (1994)

A useful brief review of standard definitions and facts about spectral sequences is

• Hal Schenck, Chapter 9: Cohomology and spectral sequences (pdf)

Nonabelian / unstable theory

Homotopy spectral sequences in model categories are discussed in

• A. Bousfield, Cosimplicial resolutions and homotopy spectral sequences in model categories (arXiv:math/0312531).

Spectral sequences in a categories with zero morphisms are discussed in

• Marco Grandis, Homotopy spectral sequences (arXiv:1007.0632)