and
nonabelian homological algebra
algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
The notion of spectral sequence is an algorithm or computational tool in homological algebra and more generally in homotopy theory which allows to compute chain homology groups/homotopy groups of bi-graded objects from the homology/homotopy of the two graded components.
Notably there is a spectral sequence for computing the homology of the total complex of a double complex from the homology of its row and column complexes separately. This in turn allows to compute derived functors of composite functors $G\circ F$ from the double complex $\mathbb{R}^\bullet G (\mathbb{R}^\bullet F(-))$ obtained by non-totally deriving the two functors separately (called the Grothendieck spectral sequence). By choosing various functors $F$ and $G$ here this gives rise to various important classes of examples of spectral sequences, see below.
More concretely, a homology spectral sequence is a sequence of graded chain complexes that provides the higher order corrections to the naïve idea of computing the homology of the total complex $Tot(V)_\bullet$ of a double complex $V_{\bullet, \bullet}$: by first computing those of the vertical differential, then those of the horizontal differential induced on these vertical homology groups (or the other way around). This simple idea in general does not produce the correct homology groups of $Tot(V)_\bullet$, but it does produce a “first-order approximation” to them, in a useful sense. The spectral sequence is the sequence of higher-order corrections that make this naive idea actually work.
Being, therefore, an iterative perturbative approximation scheme of bigraded differential objects, fully-fledged spectral sequences can look a bit intricate. However, a standard experience in mathematical practice is that for most problems of practical interest the relevant spectral sequence “perturbation series” yields the exact result already at the second stage. This reduces the computational complexity immensely and makes spectral sequences a wide-spread useful computational tool.
Despite their name, there seemed to be nothing specifically “spectral” about spectral sequences, for any of the technical meanings of the word spectrum. Together with the concept, this term was introduced by Jean Leray and has long become standard, but was never really motivated (see p. 5 of Chow). But then, by lucky coincidence it turns out in the refined context of stable (∞,1)-category theory/stable homotopy theory that spectral sequences frequently arise by considering the homotopy groups of sequences of spectra. This is discussed at spectral sequence of a filtered stable homotopy type.
While therefore spectral sequences are a notion considered in the context of homological algebra and more generally in stable homotopy theory, there is also an “unstable” or nonabelian variant of the notion in plain homotopy theory, called homotopy spectral sequence.
We give the general definition of a (co)homology spectral sequence. For motivation see the example Spectral sequence of a filtered complex below.
Throughout, let $\mathcal{A}$ be an abelian category.
A cohomology spectral sequence in $\mathcal{A}$ is
a family $(E^{p,q}_r)$ of objects in $\mathcal{A}$, for all integers $p,q,r$ with $r\geq 1$
(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)
satisfying $d_r^2 = 0$ (more precisely, $d_r^{p+r,q-r+1}\circ d_r^{p,q} = 0$)
isomorphisms $\alpha_r^{p,q}: H^{p,q}(E_r)\to E^{p,q}_{r+1}$ where the chain cohomology is given by
Analogously a homology spectral sequence is collection of objects $(E_{p,q}^r)$ with the differential $d_r$ of degree $(-r,r-1)$.
Let $\{E^r_{p,q}\}_{r,p,q}$ be a spectral sequence such that for each $p,q$ there is $r(p,q)$ such that for all $r \geq r(p,q)$ we have
Then one says equivalently that
the bigraded object
is the limit term of the spectral sequence;
the spectral sequence abuts to $E^\infty$.
If for a spectral sequence there is $r_s$ such that all differentials on pages after $r_s$ vanish, $\partial^{r \geq r_s} = 0$, then $\{E^{r_s}\}_{p,q}$ is limit term for the spectral sequence. One says in this cases that the spectral sequence degenerates at $r_s$.
By the defining relation
the spectral sequence becomes constant in $r$ from $r_s$ on if all the differentials vanish, so that $ker(\partial^r_{p,q}) = E^r_{p,q}$ for all $p,q$.
If for a spectral sequence $\{E^r_{p,q}\}_{r,p,q}$ there is $r_s \geq 2$ such that the $r_s$th page is concentrated in a single row or a single column, then the the spectral sequence degenerates on this pages, example 1, hence this page is a limit term, def. 2. One says in this case that the spectral sequence collapses on this page.
For $r \geq 2$ the differentials of the spectral sequence
have domain and codomain necessarily in different rows an columns (while for $r = 1$ both are in the same row and for $r = 0$ both coincide). Therefore if all but one row or column vanish, then all these differentials vanish.
A spectral sequence $\{E^r_{p,q}\}_{r,p,q}$ converges weakly to a graded object $H_\bullet$ with exhaustive filtering $F_\bullet H_\bullet$, traditionally denoted
if the associated graded complex $\{G_p H_{p+q}\}_{p,q} \coloneqq \{F_p H_{p+q} / F_{p-1} H_{p+q}\}$ of $H$ is the limit term of $E$, def. 2:
Furthermore one says that $E$
converges if $F_\bullet$ is also a Hausdorff filtration;
converges strongly if $F_\bullet$ is also a complete Hausdorff filtration.
(Boardman 99, def. 2.5) See also conditional convergence.
In practice, spectral sequences are often referred to via their first interesting page, usually the first or the second. Then one often uses notation such as
or
to be read as “There is a spectral sequence whose first/second page is as shown on the left and which converges (weakly, strongly, or conditionally) to a filtered object as shown on the right.”
The nature of the convergence condition in def. 3 is well illuminated for instance by the Serre-Atiyah-Hirzebruch spectral sequence: for $E^\bullet$ a generalized cohomology theory and $X$ a finite CW-complex, then it converges to the $E$-cohomology of $X$, filtered by $E$-cohomology relative to the skeleta $F^p E^{\bullet}(X) im( E^\bullet(X,X_{p-1}) \to E^\bullet(X))$. Moreover, the second page is the ordinary cohomology of $X$ with coefficients in the $E$-ground ring, like so:
Here the elements on the left in bidegree $(p,q)$ are manifestly given by cocycles that trivialize on the $(p-1)$-skeleton (being $p$-cocycles), hence it is natural that these contribute to the filtering stage $F^p E^{p+q}(X) = im(E^{p+q}(X,X_{p-1}) \to E^{p+q}(X))$.
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 $r\geq r_0$, $d_r^{p-r,q+r-1} = 0$.
If $(E^r)$ collapses at $r$, then it converges to $H_\bullet$ with $H_n$ being the unique entry $E_{p,q}^r$ on the non-vanishing row/column with $p+q = n$.
A spectral sequence $\{E^r_{p,q}\}$ is called a bounded spectral sequence if for all $n,r \in \mathbb{Z}$ the number of non-vanishing terms of the form $E^r_{k,n-k}$ is finite.
A spectral sequence $\{E^r_{p,q}\}$ is called
a first quadrant spectral sequence if all terms except possibly for $p,q \geq 0$ vanish;
a third quadrant spectral sequence if all terms except possibly for $p,q \leq 0$ vanish.
Such spectral sequences are bounded, def. 4.
First notice that if a spectral sequence has at most $N$ non-vanishing terms of total degree $n$ on page $r$, then all the following pages have at most at these positions non-vanishing terms, too, since these are the homologies of the previous terms.
Therefore for a bounded spectral sequence for each $n$ there is $L(n) \in \mathbb{Z}$ such that $E^r_{p,n-p} = 0$ for all $p \leq L(n)$ and all $r$. Similarly there is $T(n) \in \mathbb{Z}$ such $E^r_{n-q,q} = 0$ for all $q \leq T(n)$ and all $r$.
We claim then that the limit term of the bounded spectral sequence is in position $(p,q)$ given by the value $E^r_{p,q}$ for
This is because for such $r$ we have
$E^r_{p-r, q+r-1} = 0$ because $p-r \lt L(p+q-1)$, and hence the kernel $ker(\partial^r_{p-r,q+r-1}) = 0$ vanishes;
$E^r_{p+r, q-r+1} = 0$ because $q-r + 1 \lt T(p+q+1)$, and hence the image $im(\partial^r_{p,q}) = 0$ vanishes.
Therefore
Given a spectral sequence, then even if it converges strongly (def. 3), computing its infinity-page still just gives the associated graded of the filtered object that it converges to, not the filtered object itself. The latter is in each filter stage an extension of the previous stage by the corresponding stage of the infinity-page, but there are in general several possible extensions (the trivial extension or some twisted extensions). The problem of determining these extensions and hence the problem of actually determining the filtered object from a spectral sequence converging to it is often referred to as the extension problem.
More in detail, consider, for definiteness, a cohomology spectral sequence (def. 1) converging (def. 3) to some filtered $F^\bullet H^\bullet$
Then by definition of convergence there are isomorphisms
Equivalently this means that there are short exact sequences of the form
for all $p$. The extension problem then is to inductively deduce $F^p H^\bullet$ from knowledge of $F^{p+1}H^\bullet$ and $E_\infty^{p,\bullet}$.
In good cases these short exact sequences happen to be split exact sequences, which means that the extension problem is solved by the direct sum
But in general this need not be the case.
One sufficient condition that these exact sequences split is that they consist of homomorphisms of $R$-modules, for some $R$, and that $E_\infty^{p,\bullet}$ are projective modules (for instance free modules) over $R$. Because then the Ext-group $Ext^1_R(E_\infty^{p,\bullet},-)$ vanishes, and hence all extensions are trivial, hence split.
So for instance for every spectral sequence in vector spaces the extension problem is trivial (since every vector space is a free module).
The basic class of examples are
which compute the cohomology of a filtered complex from the cohomologies of its associated graded objects.
From this one obtains as a special case the class of
which compute the cohomology of the total complex of a double complex using the two canonical filtrations of this by row- and by column-degree.
From this in turn one obtains as a special case the class of
which compute the derived functor $\mathbb{R}^\bullet(G \circ F (-))$ of the composite of two functors from the spectral sequence of the double complex $\mathbb{R}^\bullet (F (\mathbb{R}^\bullet G (-)))$.
Many special cases of this for various choices of $F$ and $G$ go by special names, this we tabulate at
The fundamental example of a spectral sequence, from which essentially all the other examples arise as special cases, is the spectral sequence of a filtered complex. (See there for details). Or more generally in stable homotopy theory: the spectral sequence of a filtered stable homotopy type.
If a cochain complex $C^\bullet$ is equipped with a filtration $F^\bullet C^\bullet$, there is an induced filtration $F^\bullet H(C)$ of its cohomology groups, according to which levels of the filtration contain representatives for the various cohomology classes.
A filtration $F$ also gives rise to an associated graded object $Gr(F)$, whose grades are the successive level inclusion cokernels. Generically, the operations of grading and cohomology do not commute:
But the spectral sequence associated to a filtered complex $F^\bullet C^\bullet$, passes through $H^\bullet (Gr(F^\star) C)$ in the page $E_{(1)}$ and in good cases converges to $Gr(F^* H^\bullet(C))$.
The total complex of a double complex is naturally filtered in two ways: by columns and by rows. By the above spectral sequence of a filtered complex this gives two different spectral sequences associated computing the cohomology of a double complex from the cohomologies of its rows and columns. Many other classes of spectral sequences are special cases of this cases, notably the Grothendieck spectral sequence and its special cases.
This is discussed at spectral sequence of a double complex.
From the spectral sequence of a double complex one obtains as a special case a spectral sequence that computes hyper-derived functors.
(…)
The Grothendieck spectral sequence computes the composite of two derived functors from the two derived functors separately.
Let $\mathcal{A} \stackrel{F}{\to} \mathcal{B} \stackrel{G}{\to} \mathcal{C}$ be two left exact functors between abelian categories.
Write $R^p F : \mathcal{D} \to Ab$ for the cochain cohomology of the derived functor of $F$ in degree $p$ etc. .
If $F$ sends injective objects of $\mathcal{A}$ to $G$-acyclic objects in $\mathcal{B}$ then for each $A \in \mathcal{A}$ there is a first quadrant cohomology spectral sequence
that converges to the right derived functor of the composite functor
Moreover
the edge maps in this spectral sequence are the canonical morphisms
induced from applying $F$ to an injective resolution $A \to \hat A$ and the morphism
the exact sequence of low degree terms is
This is called the Grothendieck spectral sequence.
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 $\mathbb{R}^p(G) (F(A))$.
Therefore the second spectral sequence discussed at hyper-derived functor spectral sequences converges as
Now since by construction $H^q(F(\hat A)) = R^q F(A)$ this is a spectral sequence
This is the Grothendieck 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-forward of sheaves of 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 $Y$.
Let $X, Y$ be suitable sites and $f : X \to Y$ be a morphism of sites. () Let $\mathcal{C} = Ch_\bullet(Sh(X,Ab))$ and $\mathcal{D} = Ch_\bullet(Sh(Y,Ab))$ be the model categories of complexes of sheaves of abelian groups. The direct image $f_*$ and global section functor $\Gamma_Y$ compose to $\Gamma_X$:
Then for $A \in Sh(X,Ab)$ a sheaf of abelian groups on $X$ there is a cohomology spectral sequence
that converges as
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 $A$.
For $R$ a ring write $R$Mod for its category of modules. Given a homomorphism of ring $f : R_1 \to R_2$ and an $R_2$-module $N$ there are composites of base change along $f$ with the hom-functor and the tensor product functor
The derived functors of $Hom_{R_2}(-,N)$ and $\otimes_{R_2} N$ are the Ext- and the Tor-functors, respectively, so the Grothendieck spectral sequence applied to these composites yields base change spectral sequence for these.
The above examples are all built on the spectral sequence of a filtered complex. An alternatively universal construction builds spectral sequences from exact couples.
An exact couple is an exact sequence of three arrows among two objects
These creatures construct spectral sequences by a two-step process:
is again an exact couple.
The sequence of complexes $(E,d),(E',d'),\dots$ is a spectral sequence, by construction. For more see at exact couple – Spectral sequences from exact couples
Examples of exact couples can be constructed in a number of ways. Notably there are naturally exact couples of towers of (co-)fibrations. For instance Adams spectral sequences are usually produced this way (from towers which are Adams resolutions). For a list of examples in this class see below.
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
gives the exact couple
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 $\coker \varphi\to E\to \ker\varphi$, 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 $j,k,\varphi$, leading to the conventions mentioned earlier.
The following list of examples orders the various classes of spectral sequences by special cases: items further to the right are special cases of items further to the left.
spectral sequence of a filtered complex (approximate chain homology by higher-order relative homology in the presence of a filtering)
Atiyah-Hirzebruch spectral sequence (compute generalized homology of a Serre fibration in terms of that of the base with coefficients in that of the fibers)
Eilenberg-Moore spectral sequence (compute homology of homotopy fiber products)
spectral sequence of a double complex (compute homology of total complex by filtering by row/column degree)
Grothendieck spectral sequence (compute composition of two derived functors)
base change spectral sequence (compute base change/extension of scalars in two stages)
Leray spectral sequence (compute global sections in two stages)
Hochschild-Serre spectral sequence (compute group cohomology in two stages)
Here is a more random list (using material from Wikipedia). Eventually to be merged with the above.
Adams–Novikov spectral sequence, converging to homotopy groups of connective spectra
chromatic spectral sequence for calculating the initial terms of the Adams–Novikov spectral sequence
Atiyah–Hirzebruch spectral sequence of an extraordinary cohomology theory;
Bar spectral sequence? for the homology of the classifying space of a group
Barratt spectral sequence? converging to the homotopy of the initial space of a cofibration
Bloch–Lichtenbaum spectral sequence? converging to the algebraic K-theory of a field
Bockstein spectral sequence relating the homology with $mod p$ coefficients and the homology reduced $mod p$
Bousfield–Kan spectral sequence converging to the homotopy colimit of a functor
Cartan–Leray spectral sequence? converging to the homology of a quotient space
Cech-to-derived functor spectral sequence? from Čech cohomology to abelian sheaf cohomology
change of rings spectral sequences? for calculating Tor and Ext groups of modules
Connes spectral sequences? converging to the cyclic homology of an algebra
EHP spectral sequence converging to stable homotopy groups of spheres
Eilenberg–Moore spectral sequence for the singular cohomology of the pullback of a fibration
Federer spectral sequence? converging to homotopy groups of a function space
Frölicher spectral sequence starting from the Dolbeault cohomology and converging to the algebraic de Rham cohomology of a variety
Grothendieck spectral sequence for composing derived functors
Hodge–de Rham spectral sequence converging to the algebraic de Rham cohomology of a variety
Hurewicz spectral sequence? for calculating the homology of a space from its homotopy groups
hyperhomology spectral sequence? for calculating hyperhomology?
Künneth spectral sequence for calculating the homology of a tensor product of differential graded algebras
Leray spectral sequence converging to abelian sheaf cohomology
Lyndon–Hochschild–Serre spectral sequence? in group cohomology
May spectral sequence for calculating the Tor or Ext groups of an algebra
Miller spectral sequence? converging to the $mod p$ stable homology? of a space
Quillen spectral sequence? for calculating the homotopy of a simplicial group
van Est spectral sequence? converging to relative Lie algebra cohomology
van Kampen spectral sequence? for calculating the homotopy of a wedge of spaces.
(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 $s \geq r$.
(classical convergence theorem)
(…)
This is recalled in (Weibel, theorem 5.51).
A first quadrant spectral sequence is one for wich all pages are concentrated in the first quadrant of the $(p,q)$-plane, in that
If the $r$th page is concentrated in the first quadrant, then so the $(r+1)st$ page. So if the first one is, then all are.
Every first quadrant spectral sequence converges at $(p,q)$ from $r \gt max(p,q+1)$ on
If a first quadrant spectral sequence converges
then each $H^n$ has a filtration of length $n+1$
and we have
$F^n H^n \simeq E_\infty^{n,0}$
$H^n/F^1 H^n \simeq E_\infty^{0,n}$.
Cohomological spectral sequences are compatible with cup product structure on the $E_2$-page. (e.g. Hutchings 11, sections 5 and 6)
An elementary pedagogical introduction is in
Textbook accounts include
John McCleary, A User’s Guide to Spectral Sequences, Cambridge University Press
Charles Weibel, chapter 5, An introduction to homological algebra Cambridge studies in advanced mathematics 38 (1994)
Raoul Bott, Loring Tu, section 14 of Differential forms in algebraic topology, Graduate Texts in Mathematics 82, Springer 1982. xiv+331 pp.
Hal Schenck, Chapter 9: Cohomology and spectral sequences (pdf) .
Alan Hatcher?, Spectral sequences in algebraic topology (web)
Stanley Kochmann, Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
The general discussion in the context of stable (∞,1)-category theory (the spectral sequence of a filtered stable homotopy type) is in section 1.2.2 of
A review Master thesis is
Reviews of and lecture notes on standard definitions and facts about spectral sequences include
Matthew Greenberg, Spectral sequences (pdf)
Michael Hutchings, Introduction to spectral sequences (pdf)
Daniel Murfet, Spectral sequences (pdf)
Neil Strickland, Spectral sequences (pdf)
Ravi Vakil, Spectral Sequences: Friend or Foe? (pdf)
Brandon Williams, Spectral sequences (pdf)
Michael Boardman, Conditionally convergent spectral sequences, 1999 (pdf)
See also
A. Romero, J. Rubio, F. Sergeraert, Computing spectral sequences (pdf)
Eric Peterson, Ext chart software for computing spectral sequences
Homotopy spectral sequences in model categories are discussed in
Spectral sequences in general categories with zero morphisms are discussed in
Discussion from a perspective of homotopy type theory is in