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spectral sequence of a filtered stable homotopy type

Contents

Context

Stable Homotopy theory

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Idea

Each filtering on an object XX in a suitable stable (∞,1)-category 𝒞\mathcal{C} (a stable homotopy type XX such as a spectrum object, in particular possibly a chain complex) induces a spectral sequence whose first page consists of the homotopy groups of the homotopy cofibers of the filtering and which under suitable conditions converges to the homotopy groups of the total object XX.

This is a generalization of the traditional spectral sequence of a filtered complex to which it reduces for 𝒞=Ch (𝒜)\mathcal{C} = Ch_\bullet(\mathcal{A}) an (∞,1)-category of chain complexes presented in the projective model structure on chain complexes. Therefore the whole zoo of traditional spectral sequences that in turn are special cases of that of a filtered complex (see at spectral sequence – Examples) is all subsumed by the concept of spectral sequence of a filtered stable homotopy type.

Moreover, by applying general (∞,1)-categorical notion to naturally arising towers (such as the Whitehead tower, the chromatic tower) it naturally produces more specialized spectral sequences (such as the Atiyah-Hirzebruch spectral sequence, the chromatic spectral sequence, etc.). Specifically, applied to a coskeleton tower of a dual Cech nerve of an E-∞ algebra EE it naturally produces the EE-Adams spectral sequence. See the discussion of the Examples below.

Therefore in (∞,1)-category theory one finds a lucky coincidence of historical terminology: spectral sequences are essentially sequences of spectra, when considered on homotopy groups.

The general construction can be summarized as follows:

Any homological functor

𝒞𝒜 \mathcal{C}\to\mathcal{A}

from a stable (∞,1)-category to an abelian category induces a functor

Filt(𝒞)SpSeq(𝒜) Filt(\mathcal{C}) \to SpSeq(\mathcal{A})

from the stable (∞,1)-category of filtered objects in 𝒞\mathcal{C} to the abelian category of bigraded spectral sequences in 𝒜\mathcal{A}.

Definition

Let throughout 𝒞\mathcal{C} be a stable (∞,1)-category, 𝒜\mathcal{A} an abelian category, and π:𝒞𝒜\pi \;\colon\; \mathcal{C}\longrightarrow \mathcal{A} a homological functor on 𝒞\mathcal{C}, i.e., a functor that transforms every cofiber sequence

XYZΣX X\to Y\to Z\to \Sigma X

in 𝒞\mathcal{C} into a long exact sequence

π(X)π(Y)π(Z)π(ΣX) \dots \to \pi(X)\to \pi(Y)\to \pi(Z)\to \pi(\Sigma X) \to \dots

in 𝒜\mathcal{A}. We write π n=πΣ n\pi_n=\pi\circ \Sigma^{-n}.

Example
  • 𝒞\mathcal{C} is arbitrary, 𝒜\mathcal{A} is the category of abelian groups and π\pi is taking the 0th homotopy group π 0𝒞(S,)\pi_0 \mathcal{C}(S,-) of the mapping spectrum out of some object S𝒞S\in\mathcal{C}

  • 𝒞\mathcal{C} is equipped with a t-structure, 𝒜\mathcal{A} is the heart of the t-structure, and π\pi is the canonical functor.

  • 𝒞=D(𝒜)\mathcal{C} = D(\mathcal{A}) is the derived category of the abelian category 𝒜\mathcal{A} and π=H 0\pi=H_0 is the degree-0 chain homology functor.

  • Any of the above with 𝒞\mathcal{C} and 𝒜\mathcal{A} replaced by their opposite categories.

(Co-)Filtered objects and their (co-)chain complexes

Definition

A filtered object in an (∞,1)-category in 𝒞\mathcal{C} is an object X𝒞X \in \mathcal{C} together with a sequential diagram X:(,<)𝒞X \colon (\mathbb{Z}, \lt) \to \mathcal{C}

X n1X nX n+1 \cdots X_{n-1} \to X_n \to X_{n+1} \to \cdots

and an equivalence

Xlim nX n X \coloneqq \underset{\rightarrow}{\lim}_n X_n

between XX and the (∞,1)-colimit of this sequence. (The sequence itself is the filtering on XX.)

Dually, a co-filtered object in an (∞,1)-category in 𝒞\mathcal{C} is an object X𝒞X \in \mathcal{C} together with a sequential diagram X:(,<)𝒞X \colon (\mathbb{Z}, \lt) \to \mathcal{C}

X n1X nX n+1 \cdots X_{n-1} \to X_n \to X_{n+1} \to \cdots

and an equivalence

Xlim nX n X \coloneqq \underset{\leftarrow}{\lim}_n X_n

between XX and the (∞,1)-limit of this sequence. (The sequence itself is the co-filtering on XX.)

This appears as (Higher Algebra, def. 1.2.2.9). The notions are equivalent under replacing 𝒞\mathcal{C} by its opposite category 𝒞 op\mathcal{C}^{op}.

Remark

If 𝒞\mathcal{C} is presented by a sufficiently nice model category CC (for instance a combinatorial model category), then (∞,1)-colimits in 𝒞\mathcal{C} are computed by homotopy colimits in CC. These in turn are computed as ordinary colimits in CC over a cofibrant diagram in the projective model structure on functors.

Specifically, as discussed at homotopy limit – Examples – Over sequential diagrams a cofibrant resolution of a sequential diagram (,)C(\mathbb{N}, \leq) \to C is a sequential diagram all whose whose objects are cofibrant and all whose morphisms are cofibrations in CC

cofX 0cofX n1 CcofX n CcofX n+1 C, \emptyset \stackrel{cof}{\to} X_0 \stackrel{cof}{\to} \cdots \to X^C_{n-1} \stackrel{cof}{\to} X_{n}^C \stackrel{cof}{\to} X_{n+1}^C \to \cdots \,,

where X n CCX_n^C \in C denotes an object in the model category presenting the given object X n𝒞X_n \in \mathcal{C}.

Moreover, in many model categories that appear in practice the cofibrations are in particular monomorphisms, this is the case in particular in a projective model structure on chain complexes. In these cases then a filtering on an object X𝒞X \in \mathcal{C} in the abstract sense of (∞,1)-categories is presented by a filtered object X CCX^C \in C in the sense of plain category theory.

The intrinsic definition makes manifest however that the monomorphism-aspect here is just a means of a presentation of the filtering and not an intrinsic aspect of the homotopy theory.

Definition

Let II be a linearly ordered set. An II-chain complex in a stable (∞,1)-category 𝒞\mathcal{C} is an (∞,1)-functor

F:I Δ[1]𝒞 F \;\colon\; I^{\Delta[1]} \longrightarrow \mathcal{C}

from the subposet of I×II \times I on pairs of elements iji \leq j, such that

  1. for each nIn \in I, F(n,n)0F(n,n) \simeq 0 is the zero object;

  2. for all ijki \leq j \leq k the induced diagram

    F(i,j) F(i,k) 0 F(j,j) F(j,k) \array{ & F(i,j) &\longrightarrow& F(i,k) \\ & \downarrow && \downarrow \\ 0 \simeq & F(j,j) &\longrightarrow& F(j,k) }

    is a homotopy pushout square (hence equivalently, by stability, a homotopy pullback).

Write

Gap(I,𝒞)Func(I Δ[1],𝒞) Gap(I,\mathcal{C}) \hookrightarrow Func(I^{\Delta[1]}, \mathcal{C})

for the full sub-(∞,1)-category of diagrams satisfying these conditions.

This is Higher Algebra, def. 1.2.2.2.

Remark

The conditions in def. imply by the pasting law that also all squares

F(i,k) F(i,l) F(j,k) F(j,l) \array{ F(i,k) &\longrightarrow& F(i,l) \\ \downarrow && \downarrow \\ F(j,k) &\longrightarrow& F(j,l) }

for all iki \leq k and klk \leq l are homotopy pushout squares.

Definition

Given a \mathbb{Z}-chain complex FF in 𝒞\mathcal{C} as in def. , define a sequential diagram in the (triangulated) homotopy category Ho(𝒞)Ho(\mathcal{C}) of 𝒞\mathcal{C}

C :(,) opHo(𝒞) C_\bullet \;\colon\; (\mathbb{Z}, \leq)^{op} \longrightarrow Ho(\mathcal{C})

by setting

C nΣ nF(n1,n)Ho(𝒞) C_n \coloneqq \Sigma^{-n} F(n-1,n) \in Ho(\mathcal{C})

and taking

d nΣ nδ n:C nC n1 d_n \coloneqq \Sigma^{-n} \delta_n \;\colon\; C_n \longrightarrow C_{n-1}

to be the nn-fold de-suspension of the connecting homomorphisms of the defining homotopy fiber sequences

F(n1,n)F(n1,n+1)F(n,n+1), F(n-1,n) \to F(n-1, n+1) \to F(n,n+1) \,,

hence the (n+1)(n+1)-fold de-suspension of the morphism δ n\delta_n in the following pasting of homotopy pushouts

F(n1,n) F(n1,n+1) 0 0 F(n,n+1) δ n ΣF(n1,n) \array{ F(n-1,n) &\longrightarrow& F(n-1,n+1) &\to& 0 \\ \downarrow && \downarrow && \downarrow \\ 0 &\longrightarrow& F(n,n+1) &\stackrel{\delta_n}{\longrightarrow}& \Sigma F(n-1,n) }

where the total outer homotopy pushout exhibits the suspension of F(n1,n)F(n-1,n), by the pasting law.

Proposition

The sequence C C_\bullet in def. is a chain complex in that the d d_\bullet are differentials, hence in that for all nn \in \mathbb{Z} we have that the composite

d nd n+1=0 d_n \circ d_{n+1} = 0

is the zero morphism in the triangulated category Ho(𝒞)Ho(\mathcal{C}).

(Higher Algebra, remark 1.2.2.3)

Proof

Consider the pasting diagram

F(n2,n) F(n2,n+1) 0 (c) F(n1,n) F(n1,n+1) 0 (c) (c) 0 F(n,n+1) δ n ΣF(n1,n) \array{ F(n-2,n) &\longrightarrow& F(n-2,n+1) &\longrightarrow& 0 \\ \downarrow &(c)& \downarrow && \downarrow \\ F(n-1,n) &\longrightarrow& F(n-1,n+1) &\longrightarrow& 0 \\ \downarrow &(c)& \downarrow &(c)& \downarrow \\ 0 &\longrightarrow& F(n,n+1) &\underset{\delta_n}{\longrightarrow}& \Sigma F(n-1,n) }

where the squares labeled “c” are (co-)cartesian (homotopy pushouts) ( by def. and by remark and ). Notice that the homotopy pushout of the outermost span gives the suspension

F(n2,n) 0 (c) 0 ΣF(n2,n). \array{ F(n-2,n) &\longrightarrow& 0 \\ \downarrow &(c)& \downarrow \\ 0 &\longrightarrow& \Sigma F(n-2,n) } \,.

Therefore we have two paths of morphisms of span diagrams, the first is

(F(n2,n) F(n2,n+1) 0)(F(n2,n) 0 0)(F(n1,n) 0 0) \left( \array{ F(n-2,n) &\to& F(n-2,n+1) \\ \downarrow \\ 0 } \right) \to \left( \array{ F(n-2,n) &\to& 0 \\ \downarrow \\ 0 } \right) \to \left( \array{ F(n-1,n) &\to& 0 \\ \downarrow \\ 0 } \right)

which gives on homotopy pushouts

F(n,n+1)ΣF(n2,n)ΣF(n1,n) F(n,n+1) \longrightarrow \Sigma F(n-2,n) \longrightarrow \Sigma F(n-1,n)

and the second is

(F(n2,n) F(n2,n+1) 0)(F(n1,n) F(n1,n+1) 0)(F(n1,n) 0 0) \left( \array{ F(n-2,n) &\to& F(n-2,n+1) \\ \downarrow \\ 0 } \right) \to \left( \array{ F(n-1,n) &\to& F(n-1,n+1) \\ \downarrow \\ 0 } \right) \to \left( \array{ F(n-1,n) &\to& 0 \\ \downarrow \\ 0 } \right)

which on homotopy pushouts is

F(n,n+1)F(n,n+1)δ nΣF(n1,n) F(n,n+1) \stackrel{\simeq}{\longrightarrow} F(n,n+1) \stackrel{\delta_n}{\longrightarrow} \Sigma F(n-1,n)

(all by the pasting law). By the commutativity of the original pasting diagram these two paths are equivalent. Therefore on homotopy pushouts this exhibits a factorization of δ n\delta_n through ΣF(n2,n)\Sigma F(n-2,n):

F(n,n+1) ΣF(n2,n) δ n ΣF(n1,n). \array{ F(n,n+1) &\longrightarrow& \Sigma F(n-2,n) \\ & {}_{\mathllap{\delta_n}}\searrow & \downarrow \\ && \Sigma F(n-1,n) } \,.

Pasting this to the homotopy pushout that defines Σδ n1\Sigma \delta_{n-1}

F(n,n+1) ΣF(n2,n) 0 δ n (c) ΣF(n1,n) Σδ n1 Σ 2F(n2,n1) \array{ F(n,n+1) &\longrightarrow& \Sigma F(n-2,n) &\longrightarrow& 0 \\ & {}_{\mathllap{\delta_n}}\searrow & \downarrow &(c)& \downarrow \\ && \Sigma F(n-1,n) &\underset{\Sigma \delta_{n-1}}{\longrightarrow}& \Sigma^2 F(n-2,n-1) }

and then suspending the result nn times yields a diagram that exhibits a null-homotopy

d n1d n0 d_{n-1} \circ d_n \simeq 0

in 𝒞\mathcal{C}.

The following proposition observes that the \mathbb{Z}-chain complexes of def. are, despite the explict appearance of square diagrams, equivalently already determined by a sequential diagram.

Proposition

Consider the inclusion of posets

(,)(,) Δ[1] (\mathbb{Z}, \leq) \to (\mathbb{Z}, \leq)^{\Delta[1]}

given by

n(,n). n \mapsto (-\infty, n) \,.

The induced (∞,1)-functor

Func(({},) Δ[1],𝒞)Func((,),𝒞) Func((\mathbb{Z}\cup \{-\infty\}, \leq)^{\Delta[1]} , \mathcal{C}) \longrightarrow Func((\mathbb{Z}, \leq), \mathcal{C})

restricts to an equivalence between the (∞,1)-category Gap(,𝒞)Gap(\mathbb{Z},\mathcal{C}) of {}\mathbb{Z}\cup \{\infty\}-chain complexes in 𝒞\mathcal{C} (def. ) and that of filtered objects in 𝒞\mathcal{C} (def. ). The equivalence is given by left and right (∞,1)-Kan extension.

This is Higher Algebra, lemma 1.2.2.4.

Remark

The inverse functor can be described informally as follows:

given a filtered object X X_\bullet, the associated chain complex X(,)X(\bullet,\bullet) is given by taking each entry X(n,n+r)X(n,n+r) to be given by the homotopy cofiber of X nX n+rX_n \to X_{n+r}

X(n,n+r)=cofib(X nX n+r) X(n, n+r) = \operatorname{cofib}(X_n\to X_{n+r})

because that makes the squares

X(,n) X(,n+r) 0 X(n,n) X(n,n+r) \array{ & X(-\infty,n) &\longrightarrow& X(-\infty,n+r) \\ & \downarrow && \downarrow \\ 0 \simeq & X(n, n) &\longrightarrow& X(n,n+r) }

be homotopy pushout squares.

Spectral sequence of a filtered object

We discuss now how in the presence of sequential colimits, every filtered object induces a spectral sequence which converges to its homotopy groups, equipped with the induced filtering. The discussion for co-filtered objects is formally dual, but also spelled out below, for reference.

Remark

Let X X_\bullet be a filtered object in the sense of def. . Write X(,)X(\bullet,\bullet) for the corresponding \mathbb{Z}-complex, according to prop. . Then for all ijki \leq j \leq k there is a long exact sequence of homotopy groups in 𝒜\mathcal{A} of the form

π nX(i,j)π nX(i,k)π nX(j,k)π n1X(i,j). \cdots \to \pi_n X(i,j) \to \pi_n X(i,k) \to \pi_n X(j,k) \to \pi_{n-1}X(i,j) \to \cdots \,.
Definition

Define for p,qp,q \in \mathbb{Z} and r1r \geq 1 an object E r p,q𝒜E_r^{p,q} \in \mathcal{A} by the canonical epi-mono factorization

π p+qX(pr,p)E r p,qπ p+qX(p1,p+r1) \pi_{p+q} X(p-r,p) \twoheadrightarrow E_r^{p,q} \hookrightarrow \pi_{p+q} X(p-1, p+r-1)

in the abelian category 𝒜\mathcal{A}, of the morphism X((pr,p)(p1,p+r1))X((p-r,p) \leq (p-1,p+r-1)), so that E r p,qE_r^{p,q} is the image of this morphism. Moreover, define morphisms

d r:E r p,qE r pr,q+r1 d_r \;\colon\; E_r^{p,q} \to E_r^{p-r, q+r-1}

to be the restriction (the image on morphisms) of the connecting homomorphism

π p+qX(pr,p) E r p,q π p+qX(p1,p+r1) δ d r δ π p+q1X(p2r,pr) E r pr,q+r1 π p+q1X(pr1,p1) \array{ \pi_{p+q} X(p-r, p) &\longrightarrow& E_r^{p,q} &\longrightarrow& \pi_{p+q} X(p-1, p+r-1) \\ {}^{\mathllap{\delta}}\downarrow && \downarrow^{\mathrlap{d_r}} && \downarrow^{\mathrlap{\delta}} \\ \pi_{p+q-1} X(p-2r, p-r) &\longrightarrow& E_r^{p-r,q+r-1} &\longrightarrow& \pi_{p+q-1} X(p-r-1, p-1) }

in the long exact sequence of homotopy groups of remark ,

  • on the left for the case (ijk)=(p2rprp)(i \leq j \leq k) = (p-2r \leq p - r \leq p)

  • on the right for the case (ijk)=(pr1p1p+r1)(i \leq j \leq k) = (p - r - 1 \leq p - 1 \leq p + r - 1).

Remark

For r=1r = 1 def. reduces to

E 1 p,q π p+qX(p1,p) π q(Σ pX(p1,p)) π q(C p) \begin{aligned} E_1^{p,q} & \simeq \pi_{p+q} X(p-1,p) \\ & \simeq \pi_{q} (\Sigma^{-p} X(p-1,p)) \\ & \simeq \pi_{q} (C_{p}) \end{aligned}

where C pC_p is the ppth element in the chain complex associated with X(,)X(\bullet,\bullet) according to def. .

(Higher Algebra, construction 1.2.2.6)

Proposition

In def. we have d rd r=0d^r\circ d^r = 0 for all r1r \geq 1 and all p,qp,q \in \mathbb{Z}.

Moreover, there are natural isomorphisms (natural in X X_\bullet)

E r+1 p,qker(d r:E r p,qE r pr,q+r1)im(d r:E r p+r,qr+1E r p,q). E_{r+1}^{p,q} \simeq \frac{ ker(d_r \colon E_r^{p,q} \to E_r^{p-r, q+r-1}) }{ im(d_r \colon E_r^{p+r, q-r+1} \to E_r^{p,q}) } \,.

Thus, {E r ,} r1\{E_r^{\bullet,\bullet}\}_{r\geq 1} is a homology spectral sequence in the abelian category 𝒜\mathcal{A}, functorial in the filtered object X X_\bullet, with first page

E 1 p,q =π p+qcofib(X p1X p) π q(C p). \begin{aligned} E_1^{p,q} &= \pi_{p+q} \operatorname{cofib}(X_{p-1}\to X_{p}) \\ & \simeq \pi_q (C_p) \end{aligned} \,.

(Higher Algebra, prop. 1.2.2.7)

Proof

Since d rd_r is by definition the image morphism of a connecting homomorphism, for showing d rd r=0d_r \circ d_r = 0 it suffices to show that the connecting homomorphisms compose to the zero morphism, δ rδ r0\delta_r \circ \delta_r \simeq 0. This is the same argument as in the proof of prop. , generalized from vertical steps of length 1 to vertical steps of length rr.

Explicitly, we have the pasting diagram

F(p2r,p) F(p2r,p+1) 0 (c) F(pr,n) F(pr,p+1) 0 (c) (c) 0 F(p,p+r) δ r ΣF(pr,p) \array{ F(p-2r,p) &\longrightarrow& F(p-2r,p+1) &\longrightarrow& 0 \\ \downarrow &(c)& \downarrow && \downarrow \\ F(p-r,n) &\longrightarrow& F(p-r,p+1) &\longrightarrow& 0 \\ \downarrow &(c)& \downarrow &(c)& \downarrow \\ 0 &\longrightarrow& F(p,p+r) &\underset{\delta_r}{\longrightarrow}& \Sigma F(p-r,p) }

where the squares labeled “c” are (co-)cartesian (homotopy pushouts). By the universal property of the pushout, this induces a factorization

F(p,p+1) ΣF(p2r,p) δ r ΣF(pr,p). \array{ F(p,p+1) &\longrightarrow& \Sigma F(p-2r,p) \\ & {}_{\mathllap{\delta_r}}\searrow & \downarrow \\ && \Sigma F(p-r,p) } \,.

Pasting this in turn to the homotopy pushout that defines Σδ pr\Sigma \delta_{p-r}

F(p,p+1) ΣF(p2r,p) 0 δ r (c) ΣF(pr,p) Σδ r Σ 2F(p2r,p1) \array{ F(p,p+1) &\longrightarrow& \Sigma F(p-2r,p) &\longrightarrow& 0 \\ & {}_{\mathllap{\delta_r}}\searrow & \downarrow &(c)& \downarrow \\ && \Sigma F(p-r,p) &\underset{\Sigma \delta_{r}}{\longrightarrow}& \Sigma^2 F(p-2r,p-1) }

and then suspending the result nn times yields a diagram that exhibits a null-homotopy

δ rδ r0 \delta_{r} \circ \delta_r \simeq 0

in 𝒞\mathcal{C}.

Next, to show the homology isomorphisms; consider for fixed p,q,rp,q,r the usual abbreviation

CE r p,q C \coloneqq E_r^{p,q}

for the rr-relative chains,

Zker(d r:E r p,qE r pr,q+r1) Z \coloneqq ker(d_r \colon E_r^{p,q} \to E_r^{p-r, q+r-1})

for the rr-relative cycles and

Bim(d r:E r p+r,qr+1E r p,q) B \coloneqq im(d_r \colon E_r^{p+r, q-r+1} \to E_r^{p,q})

for the rr-relative boundaries, all in bidegree p,qp,q.

We claim that the canonical maps induce a sequence of morphisms in 𝒜\mathcal{A} of the form

π p+qX(pr1,p)ϕZϕZ/BψC/Bψπ p+qX(p1,p+r) \pi_{p+q} X(p-r-1, p) \stackrel{\phi}{\to} Z \stackrel{\phi'}{\to} Z/B \stackrel{\psi'}{\to} C/B \stackrel{\psi}{\to} \pi_{p+q} X(p-1, p+r)

and that ϕϕ\phi'\circ \phi is an epimorphism and ψϕ\psi \circ \phi' is a monomorphism. By the uniqueness of the image factorization in the abelian category 𝒜\mathcal{A}, this will prove the proposition.

To see that π p+qX(pr1,p)\pi_{p+q} X(p-r-1,p) is indeed in the kernel of d rd_r consider the commuting diagram

π p+qX(pr1,p) π p+q1X(p2r,pr1) π p+qX(pr,p) π p+q1X(p2r,pr) E r p,q d r E r pr,q+r1 π p+q1X(p2r,pr) π p+q1X(pr1,p1). \array{ \pi_{p+q} X(p-r-1,p) &\longrightarrow& \pi_{p+q-1} X(p-2r, p- r-1) \\ \downarrow && \downarrow & \searrow \\ \pi_{p+q} X(p-r, p) &\longrightarrow& \pi_{p+q-1}X(p-2r, p-r) \\ \downarrow && \downarrow \\ E_r^{p,q} &\stackrel{d_r}{\longrightarrow}& E_r^{p-r, q+r-1} && \pi_{p+q-1} X(p - 2r, p-r) \\ && \downarrow & \swarrow \\ && \pi_{p+q-1} X(p - r - 1, p - 1) } \,.

Since the bottom right morphism is a monomorphism by construction, the claim is equivalently that the total composite from top-left to bottom right is zero. By commutativity of the diagram this factors through the composite from top-right to bottom-right. As indicated, this in turn factors through two consecutive morphisms of an (ijk)(i \leq j \leq k)-square, which by definition of \mathbb{Z}-chain complex is null-homotopic.

By a dual argument one has that π p+qX(p1,p+r)\pi_{p+q}X(p-1, p+r) is in the coimage of d rd_r. This shows that we indeed have the above sequence of morphisms ϕϕψψ\stackrel{\phi}{\to}\stackrel{\phi'}{\to}\stackrel{\psi'}{\to}\stackrel{\psi}{\to}.

It now remains to show that ϕ\phi is an epimorphism (dually ψ\psi will be a monomorphism.) (…Higher Algebra, p. 41…)

We can now consider the convergence of the spectral sequence of prop. . To state that efficiently, first consider the following definition

Definition

Given a filtered object, def. , Xlim nX n𝒞X \simeq \underset{\longrightarrow}{\lim}_n X_n \in \mathcal{C}, say that the induced filtering on its homotopy groups F π XF^\bullet \pi_\bullet X is given by the images of the homotopy groups of the strata of XX

F pπ p+qXim(π p+qX pπ p+qX)𝒜. F^p \pi_{p+q}X \coloneqq im\left( \pi_{p+q} X_{p} \to \pi_{p+q} X \right) \,\,\, \in \mathcal{A} \,.

(Higher Algebra, p. 43)

Proposition

Assume that 𝒞\mathcal{C} admits all sequential colimits and that π\pi preserves these. Let Xlim nX nX \simeq \underset{\longrightarrow}{\lim}_n X_n be a filtered object, def. , for filtering with X n<00X_{n \lt 0} \simeq 0. Then the spectral sequence of prop. , converges to the homotopy groups of XX

E 1 p,q=π p+qcofib(X p1X p)π q(C p)π p+qX, E_1^{p,q} = \pi_{p+q} \operatorname{cofib}(X_{p-1}\to X_{p}) \simeq \pi_q (C_p) \;\;\Rightarrow\;\; \pi_{p+q} X \,,

where the first page is identified following remark .

In detail, for all p,qp,q \in \mathbb{Z} the differentials d r:E r p,qE r pr,q+r1d_r \colon E_r^{p,q} \to E_r^{p-r, q+r-1} vanish for r>pr \gt p, and the colimit (in 𝒜\mathcal{A})

E p,qlim r>pE r E^{p,q}_\infty \coloneqq \underset{\longrightarrow}{\lim}_{r \gt p} E_r

is isomorphic to the associated graded object of the filtered homotopy groups of def. :

E p,qF pπ p+q(X)/F p1π p+q(X). E^{p,q}_\infty \simeq F^p \pi_{p+q}(X) / F^{p-1} \pi_{p+q}(X) \,.

This is due to (Higher Algebra, prop. 1.2.2.14). A quick review is in (Wilson 13, theorem 1.2.1).

Proof

The assumption X n<00X_{n \lt 0} \simeq 0 implies that for i,j<0i,j \lt 0 we have, by remark ,

X(i,j)cofib(X iX j)0fori,j<0 X(i,j) \simeq cofib(X_i \to X_j) \simeq 0 \;\;\;\; for\, i,j \lt 0

and therefore it follows that E r pr,q+r1E_r^{p-r,q+r-1}, being a quotient of π p+qX(p2r,pr)\pi_{p+q} X(p-2r, p-r), vanishes for r>pr \gt p.

The same assumption implies that

X(pr,p)X pforp>r X(p-r,p) \simeq X_p \;\;\;\; for\, p \gt r

and so E p,qE_\infty^{p,q} is

E p,qim(π p+qX pπ p+qY) E_\infty^{p,q} \simeq im\left( \pi_{p+q} X_p \to \pi_{p+q} Y \right)

for

Ylim rX(p1,p+r). Y \coloneqq \underset{\longrightarrow}{\lim}_r X(p-1,p+r) \,.

We need to show that this image is the associated graded object of the filtered homotopy groups.

To that end, observe that the homotopy fiber sequences

X p1X p+rX(p1,p+r) X_{p-1} \to X_{p+r} \to X(p-1,p+r)

for all rr give a homotopy fiber sequence under the colimit over rr of the form

X p1XY. X_{p-1} \to X \to Y \,.

The corresponding long exact sequence of homotopy groups truncates on the left to read

0F p1π p+q(X)ker(f)π p+qXfπ p+qY. 0 \to F^{p-1} \pi_{p+q}(X) \stackrel{ker(f')}{\hookrightarrow} \pi_{p+q} X \stackrel{f'}{\to} \pi_{p+q}Y \,.

By construction the morphism ff' appearing here factors the morphism ff whose image we need to compute as

π p+qX g f π p+qX(p) f π p+qY \array{ && \pi_{p+q}X \\ & {}^{\mathllap{g}}\nearrow && \searrow^{\mathrlap{f'}} \\ \pi_{p+q} X(p) && \stackrel{f}{\longrightarrow} && \pi_{p+q} Y }

Using these relation we can now express E p,qim(f)E_\infty^{p,q} \simeq im(f) as:

E p,q im(f) im(f| im(g)) im(f| F pπ p+qX) F pπ p+qX/ker(f) F pπ p+qX/F p1π p+qX. \begin{aligned} E_\infty^{p,q} & \simeq im(f) \\ & \simeq im(f'|_{im(g)}) \\ & \simeq im(f'|_{F^p \pi_{p+q} X}) \\ & \simeq F^p \pi_{p+q}X/ker(f') \\ & \simeq F^p \pi_{p+q} X/F^{p-1} \pi_{p+q}X \end{aligned} \,.
Remark

While historically the appearances of the root “spectr-” in “spectral sequence” and in “spectrum” (stable homotopy types) are unrelated, prop. and prop. say that there is a lucky coincidence of terminology:

Every sequence of spectra manifests itself on homotopy groups in a spectral sequence.

Moreover, the discussion below in Examples shows that also conversely, essentially every spectral sequence that appears in practice comes from a sequence of spectra this way.

(See also the title of (Wilson 13)).

Remark

The spectral sequence above itself only actually depends to the triangulated homotopy category Ho(𝒞)Ho(\mathcal{C}). But its \infty-functorial dependence on the filtered object needs the full structure of the (∞,1)-category 𝒞\mathcal{C}

Spectral sequence of a cofiltered object

We discuss here the dual notion to the spectral sequence of a filtered object above, now for a cofiltered object.

The following does not just dualize but also change the indexing convention on top of dualization. Needs further discussion/harmonization.

Proposition

Consider the inclusion of posets

(,)({},) Δ[1] (\mathbb{Z}, \leq) \to (\mathbb{Z}\cup \{\infty\}, \leq)^{\Delta[1]}

given by

n(n,). n \mapsto (n,\infty) \,.

The induced (∞,1)-functor

Func(({},) Δ[1],𝒞)Func((,),𝒞) Func((\mathbb{Z}\cup \{\infty\}, \leq)^{\Delta[1]}, \mathcal{C}) \longrightarrow Func((\mathbb{Z}, \leq), \mathcal{C})

restricts to an equivalence between the (∞,1)-category of {}\mathbb{Z}\cup \{\infty\}-chain complexes in 𝒞\mathcal{C} (def. ) and that of generalized filtered objects in 𝒞\mathcal{C} (def. ).

Given a filtered object X X_\bullet, the associated chain complex X(,)X(\bullet,\bullet) is given by the homotopy fiber

X(n,n+r)=fib(X nX n+r). X(n, n+r) = \operatorname{fib}(X_n\to X_{n+r}).
Definition

For a cofiltered object X X_\bullet, def. , write

K nfib(X nX n+1) K_n \coloneqq fib(X_n \to X_{n+1})

for the homotopy fiber of the nnth structure map, for all nn \in \mathbb{Z}, and define an exact couple

π (K ) π (X ) π (X ) \array{ && \pi_\bullet(K_\bullet) \\ & \swarrow && \nwarrow \\ \pi_\bullet(X_\bullet) && \stackrel{}{\longrightarrow} && \pi_\bullet(X_\bullet) }

where the maps are given by the long exact sequences

π (X n+1)π (K n)π (X n)π (X n+1)π +1(K n) \cdots \to \pi_\bullet(X_{n+1}) \to \pi_\bullet(K_n) \to \pi_\bullet(X_n) \to \pi_\bullet(X_{n+1}) \to \pi_{\bullet+1}(K_n) \to \cdots

This exact couple gives rise in the usual way to a spectral sequence.

Let X X_\bullet be a cofiltered object.

Definition

Define for p,qp,q \in \mathbb{Z} and r1r \geq 1 the object E p,q rE^r_{p,q} by the canonical epi-mono factorization

π pX(qr+1,q+1)E p,q rπ pX(q,q+r) \pi_{p} X(q-r+1,q+1) \twoheadrightarrow E^r_{p,q} \hookrightarrow \pi_{p} X(q, q+r)

in the abelian category 𝒜\mathcal{A}, and define the differential

d r:E p,q rE p1,qr r d^r \;\colon\; E_{p,q}^r \to E_{p-1, q-r}^r

to be the restriction of the connecting homomorphism

π pX(q,q+r)π p1X(qr,q) \pi_{p} X(q,q+r) \to \pi_{p-1} X(q-r, q)

from the long exact sequence of remark ,
for the case i=qri=q-r, j=qj=q, and k=q+rk=q+r.

Proposition

d rd r=0d^r\circ d^r = 0 and there are natural (in X X_\bullet) isomorphisms

E r+1ker(d r)/im(d r). E^{r+1}\cong \operatorname{ker}(d^r)/\operatorname{im}(d^r).

Thus, {E *,* r} r1\{E^r_{*,*}\}_{r\geq 1} is a bigraded spectral sequence in the abelian category 𝒜\mathcal{A}, functorial in the filtered object X X_\bullet, with

E p,q 1=π pfib(X qX q+1),d r:E p,q rE p1,qr r. E^1_{p,q} = \pi_p \operatorname{fib}(X_q\to X_{q+1}), \qquad d^r: E^r_{p,q}\to E^r_{p-1,q-r}.

If sequential limits and sequential colimits exist in 𝒜\mathcal{A}, we can form the limiting term E *,* E^\infty_{*,*} of this spectral sequence.

On the other hand, the graded object π (X)\pi_\bullet (X) admits a filtration by

F qπ p(X)=ker(π p(X)π p(X q)) F_q \pi_p (X) = \operatorname{ker}(\pi_p (X)\to \pi_p(X_q))

and we would like to compare E *,* E^\infty_{*,*} with the associated graded of this filtration. We say that

Definition

The spectral sequence converges weakly if there is a canonical isomorphism

E p,q F qπ p(X)/F q1π p(X) E^\infty_{p,q} \cong F_q\pi_p(X)/ F_{q-1}\pi_p(X)

for every p,qp,q\in\mathbb{Z}.

We say that the spectral sequence converges strongly if it converges weakly and if, in addition, the filtration F π p(X)F_\bullet\pi_p(X) is complete on both sides, that is:

lim qF qπ p(X)π p(X)lim qF qπ p(X), \underset{\rightarrow}{\lim}_q F_q\pi_p (X) \simeq \pi_p(X) \simeq \underset{\leftarrow}{\lim}_q F^q\pi_p (X),

where F F^\bullet is the cofiltration.

Remark

The meaning of the word canonical in def. is somewhat subtle since, in general, there is no map from one side to the other. However, there always exists a canonical relation between the two, and we ask that this relation be an isomorphism (see Hilton-Stammbach, VIII.7).

Proposition

Let 𝒞\mathcal{C} be a stable (∞,1)-category and let π:𝒞𝒜\pi:\mathcal{C}\to\mathcal{A} be a homological functor where 𝒜\mathcal{A} is an abelian category which admits sequential limits. Let X X_\bullet be a filtered object in 𝒞\mathcal{C} such that limX \underset{\leftarrow}{\lim} X_\bullet exists. Suppose further that:

  1. For every nn, the diagram rfib(X nrX n)r\mapsto \operatorname{fib}(X_{n-r}\to X_n) has a limit in 𝒞\mathcal{C} and that limit is preserved by π\pi.
  2. For every nn, π n(X r)=0\pi_n(X_r)=0 for r0r\gg 0.

Then the spectral sequence {E *,* r} r1\{E^r_{*,*}\}_{r\geq 1} in 𝒜\mathcal{A} converges strongly (def. ). We write:

E p,q 1=π pfib(X qX q+1)π p(limX ) E_{p,q}^1 = \pi_{p} \operatorname{fib}(X_q\to X_{q+1}) \Rightarrow \pi_{p} (\underset{\leftarrow}{\lim} X_\bullet)

There is also a dual statement in which limits are replaced by colimits, but it is in fact a special case of the proposition with π\pi replaced by π op\pi^{op}. A proof of this proposition (in dual form) is given in (Higher Algebra, prop. 1.2.2.14). Review is in (Wilson 13, theorem 1.2.1).

Examples

General

tower diagram/filteringspectral sequence of a filtered stable homotopy type
filtered chain complexspectral sequence of a filtered complex
Postnikov towerAtiyah-Hirzebruch spectral sequence
chromatic towerchromatic spectral sequence
skeleta of simplicial objectspectral sequence of a simplicial stable homotopy type
skeleta of Sweedler coring of E-∞ algebraAdams spectral sequence
filtration by support
slice filtrationslice spectral sequence
Example

For 𝒜\mathcal{A} a good abelian category and 𝒞=Ch (𝒜)\mathcal{C} = Ch_\bullet(\mathcal{A}) the (∞,1)-category of chain complexes in 𝒜\mathcal{A}, we recover, by , the traditional notion of a spectral sequence of a filtered complex.

(Higher Algebra, example 1.2.2.11).

Example

Let 𝒞=Spec op\mathcal{C} = Spec^{op} be the opposite (∞,1)-category of spectra, let 𝒜\mathcal{A} be the opposite category of abelian groups, and let π\pi be the functor [K,][K,-] where KK is spectrum. Then condition (1) in Proposition holds for all filtered objects if and only if KK is a finite spectrum. When the filtered object is the Whitehead tower of a spectrum EE, the associated spectral sequence is the Atiyah-Hirzebruch spectral sequence with target E *(K)E^*(K). It is thus strongly convergent if KK is a finite spectrum.

Example

For 𝒞\mathcal{C} a stable (∞,1)-category and X X_\bullet a simplicial object in an (∞,1)-category in 𝒞\mathcal{C}, then the simplicial skeleta of XX give it the structure of a filtered object in an (∞,1)-category. The corresponding spectral sequence of a filtered stable homotopy type has as its first page the Moore complexes of the corresponding simplicial objects of homotopy groups.

See at spectral sequence of a simplicial stable homotopy type.

As a special case of example we have:

Example

The EE-based Adams spectral sequence that approximates homotopy classes of maps between two spectra XX and YY using a ring spectrum EE is a special case of the above spectral sequence, with 𝒞=Spec\mathcal{C}=Spec, π=[X,]\pi=[X,-], and the filtered object associated with the cosimplicial spectrum E +1YE^{\wedge\bullet+1}\wedge Y. Bousfield’s theorems on the convergence of the Adams spectral sequence can be rephrased as giving sufficient conditions on XX, YY, and EE for condition (1) in Proposition to hold (see Bousfield, Theorems 6.6 and 6.10).

See J-homomorphism and chromatic homotopy for an exposition.

Canonical cosimplicial resolution of E E_\infty-algebras

We discuss now the special case of coskeletally filtered totalizations coming from the canonical cosimplicial objects induced from E-∞ algebras (dual Cech nerves/Sweedler corings/Amitsur complexes).

In this form this appears as (Lurie 10, theorem 2). A review is in (Wilson 13, 1.3). For the analog of this in the traditional formulation see (Ravenel, ch. 3, prop. 3.1.2).

Definition

Given an cosimplicial object in an (∞,1)-category with (∞,1)-colimits

Y:Δ𝒞 Y \;\colon\; \Delta \longrightarrow \mathcal{C}

its totalization TotYlim nY nTot Y \simeq \underset{\leftarrow}{\lim}_n Y_n is filtered, def. , by the totalizations of its coskeleta

TotYTot(cosk 2Y)Tot(cosk 1Y)Tot(cosk 0Y)0. Tot Y \to \cdots \to Tot (cosk_2 Y) \to Tot (cosk_1 Y) \to Tot (cosk_0 Y) \to 0 \,.
Definition

The filtration spectral sequence, prop. , applied to the filtration of a totalization by coskeleta as in def. , we call the spectral sequence of a simplicial stable homotopy type.

(Higher Algebra, prop. 1.2.4.5)

Proposition

The spectral sequence of a simplicial stable homotopy type has as first page/E 1E_1-term the cohomology groups of the Moore complex associated with the cosimplicial objects of homotopy groups

E 2 p,q=H p(π q(Tot(cosk (Y))))π pqTot(Y). E_2^{p,q} = H^p(\pi_q(Tot (cosk_\bullet(Y)))) \Rightarrow \pi_{p-q} Tot(Y) \,.

By the discussion at ∞-Dold-Kan correspondence and spectral sequence of a filtered stable homotopy type. This appears as (Higher Algebra, remark 1.2.4.4). Review is around (Wilson 13, theorem 1.2.4).

Definition

Let SS be an E-∞ ring and let EE be an E-∞ algebra over SS, hence an E-∞ ring equipped with a homomorphism

SE. S \longrightarrow E \,.

The canonical cosimplicial object associated to this (the \infty-Cech nerve/Sweedler coring/Amitsur complex) is that given by the iterated smash product/tensor product over SS:

E S +1:Δ𝒞. E^{\wedge^{\bullet+1}_S} \;\colon\; \Delta \to \mathcal{C} \,.

More generally, for XX an SS-∞-module, the canonical cosimplicial object is

E S +1 SX:Δ𝒞. E^{\wedge^{\bullet+1}_S}\wedge_S X \;\colon\; \Delta \to \mathcal{C} \,.
Proposition

If EE is such that the self-generalized homology E (E)π (E SE)E_\bullet(E) \coloneqq \pi_\bullet(E \wedge_S E) (the dual EE-Steenrod operations) is such that as a module over E π (E)E_\bullet \coloneqq \pi_\bullet(E) it is a flat module, then there is a natural equivalence

π (E S n+1 SX)E (E S n) E E (X). \pi_\bullet \left( E^{\wedge^{n+1}_S} \wedge_S X \right) \simeq E_\bullet(E^{\wedge^n_S}) \otimes_{E_\bullet} E_\bullet(X) \,.

Reviewed for instance as (Wilson 13, prop. 1.3.1).

Remark

This makes (E ,E (E))(E_\bullet, E_\bullet(E)) be the commutative Hopf algebroid formed by the EE-Steenrod algebra. See there for more on this.

Example

The condition in prop. is satisfied for

It is NOT satisfied for

Remark

Under good conditions (…), π \pi_\bullet of the canonical cosimplicial object provides a resolution of comodule tensor product and hence computes the Ext-groups over the commutative Hopf algebroid:

H p(π q(Tot(cosk (E S +1 SX))))Ext E (E) p(Σ qE ,E (X)). H^p(\pi_q(Tot(cosk_\bullet(E^{\wedge^{\bullet+1}_S } \wedge_S X)))) \simeq Ext^p_{E_\bullet(E)}(\Sigma^q E_\bullet, E_\bullet(X)) \,.

(…)

Remark

There is a canonical map

L EXlim n(E S n+1 SX) L_E X \stackrel{}{\longrightarrow} \underset{\leftarrow}{\lim}_n (E^{\wedge^{n+1}_S}\wedge_S X)

from the EE-Bousfield localization of spectra of XX into the totalization.

(Lurie 10, lecture 30, prop. 1)

We consider now conditions for this morphism to be an equivalence.

Definition

For RR a ring, its core cRc R is the equalizer in

cRRRR. c R \longrightarrow R \stackrel{\longrightarrow}{\longrightarrow} R \otimes R \,.
Proposition

Let EE be a connective E-∞ ring such that the core or π 0(E)\pi_0(E), def. is either of

  • the localization of the integers at a set JJ of primes, cπ 0(E)matbbZ[J 1]c \pi_0(E) \simeq \matbb{Z}[J^{-1}];

  • n\mathbb{Z}_n for n2n \geq 2.

Then the map in remark is an equivalence

L EXlim n(E S n+1 SX). L_E X \stackrel{\simeq}{\longrightarrow} \underset{\leftarrow}{\lim}_n (E^{\wedge^{n+1}_S}\wedge_S X) \,.

(Bousfield 79, Lurie 10, lecture 30, prop. 3, Lurie 10, lecture 31,).

References

The general theory is set up in section 1.2.2 of

A quick exposition of that is for instance in section 1.2 of

The case of the derived category of an arbitrary abelian category is discussed in details in Chapter VIII of

  • P. Hilton, U. Stammbach, A Course in Homological Algebra, Graduate Texts in Mathematics 4

The traditional discussion of the Adams spectral sequence in this style originates in

see also at Bousfield localization of spectra. The formulation of this in modern chromatic homotopy theory is discussed in

Last revised on July 23, 2016 at 07:17:12. See the history of this page for a list of all contributions to it.