Conditionally Convergent Spectral Sequences

by J. Michael Boardman March 1999

Abstract Convergence criteria for spectral sequences are developed that apply more

widely than the traditional concepts. In the presence of additional conditions that depend on data internal to the spectral sequence, they lead to satisfactory convergence and comparison theorems. The techniques apply to whole-plane as well as half-plane spectral sequences.

Contents

Introduction 1 0 Spectral sequences 2

Part I TOOLS 4 1 Limits and colimits 4 2 Filtered groups 6 3 Image subsequences 9 4 Homotopy limits and colimits of spectra 12

Part II CONVERGENCE 15 5 Types of convergence 16 6 Half-plane spectral sequences with exiting differentials 19 7 Half-plane spectral sequences with entering differentials 20 8 Whole-plane spectral sequences 23

Part III EXAMPLES 28 9 Filtered complexes 28 10 Double complexes 29 11 Multicomplexes 30 12 Atiyah–Hirzebruch spectral sequences 31 13 Serre spectral sequences of a fibration 33 14 Bockstein spectral sequences 34 15 Adams spectral sequences 36 References 39

Introduction

§0 Spectral sequences 2

By popular demand, this paper presents material that has long circulated in preprint form, along with some newer results. Historically, the convergence of spectral sequences was handled by imposing severe finiteness conditions; this was adequate for early applications, when spectral sequences occupied only the first or third quadrant of the plane and all filtrations of groups were finite in each degree. For more general spectral sequences, such as those that fill a half-plane, Cartan and Eilenberg [4, Chap. XV] replaced finiteness conditions by limit conditions. Today, there are spectral sequences of interest that fill the whole plane; for these, better methods are essential.

The principle here, suggested at least by C. T. C. Wall and D. B. A. Epstein, is that because the limit functor of a sequence of abelian groups fails to be exact, whenever one encounters a limit group, one should for consistency take account of the associated derived limit group, which was introduced to topologists by Milnor [12]. This leads naturally to classes of spectral sequences that may or may not converge in any ordinary sense; nevertheless, one can often deduce strong convergence from data purely internal to the spectral sequence. Further, there are comparison theorems that remain valid when even weak convergence fails.

We begin by reviewing in §0 the construction of a spectral sequence. Part I introduces various algebraic and topological tools. Part II discusses convergence in great generality; as such generality is usually excessive, we include the significantly simpler cases of half-plane spectral sequences. Part III presents some common spectral sequences that automatically qualify, regardless of finiteness. No attempt is made to be comprehensive, as in McCleary’s guide [11].

0 Spectral sequences

We consider only spectral sequences of abelian groups. The extra structure that is often present is largely irrelevant to our purposes.

Unrolled exact couples All the spectral sequences we discuss arise from an unrolled (previously, unraveled) exact couple, by which we mean a diagram of graded abelian groups and homomorphisms of the form

[figure: …; A^{s+2}; A^{s+1}; A^{s}; A^{s−1}; …; …; E^{s+1}; E^{s}; E^{s−1}; …; i; j; i; j; i; j; k; k; k]  (0.1)

in which each triangle → A^{s+1} → A^{s} → E^{s} → A^{s+1} → is a long exact sequence. Each A^{s} and E^{s} is itself a graded (or occasionally bigraded) group; but we generally suppress at least one grading by working in the category of graded (or bigraded) groups. Typically (but not always), deg(i) = 0, while deg(j) and deg(k) are 0 and 1 in either order; however, the indexing and grading are often changed for reasons of convenience or personal preference to suit the application, and do not concern us.

From diagram (0.1) we extract the short exact sequence

0 ⟶ Coker[i: A^{s+1} → A^{s}] ⟶^{j} E^{s} ⟶^{k} Ker[i: A^{s+1} → A^{s}] ⟶ 0,

which shows that each E^{s} is determined up to group extension by the sequence

… ⟶ A^{s+2} ⟶^{i} A^{s+1} ⟶^{i} A^{s} ⟶^{i} A^{s−1} ⟶ …  (0.2)

of graded groups. We find that the particular extension has little relevance here; what we really need to study is the sequence (0.2), which is much simpler than (0.1). Moreover, at this point, we can discard the grading on each A^{s} and work degreewise (with adjustments if deg(i) ≠ 0).

Construction of the spectral sequence In the unrolled exact couple (0.1), the groups E^{s} are the components of the initial E_{1}-term of the spectral sequence, E_{1}^{s} = E^{s}. For the higher terms, we define for all integers s and all r ≥ 1:

Z_{r}^{s} = k^{−1}(Im[i^{(r−1)}: A^{s+r} → A^{s+1}]), the r-th cycle subgroup of E^{s} = E_{1}^{s};
B_{r}^{s} = j Ker[i^{(r−1)}: A^{s} → A^{s−r+1}], the r-th boundary subgroup of E^{s} = E_{1}^{s};
E_{r}^{s} = Z_{r}^{s}/B_{r}^{s}, a component of the E_{r}-term;

where i^{(r−1)} denotes the (r−1)-fold iterate of i. Each of these is itself a graded group, often written with components E_{r}^{s,t} etc. in degree s + t. (We warn that our indexing of Z_{r}^{s} and B_{r}^{s} is not universally accepted, e. g. MacLane [9, Chap. 11].) We thus have the subgroups of E^{s},

0 = B_{1}^{s} ⊂ B_{2}^{s} ⊂ B_{3}^{s} ⊂ … ⊂ Im j = Ker k ⊂ … ⊂ Z_{3}^{s} ⊂ Z_{2}^{s} ⊂ Z_{1}^{s} = E^{s}.

To study these, we introduce the notation

Im^{r} A^{s} = Im[i^{(r)}: A^{s+r} → A^{s}].  (0.3)

From the portion

[figure: A^{s+r+1}; A^{s+r}; A^{s+1}; A^{s}; E^{s+r}; E^{s}; i; i^{(r−1)}; j; i; j; k; k]

of diagram (0.1), we extract the two short exact sequences

0 ⟶ \frac{Z_{r}^{s}}{Ker k} ⟶^{k} Im^{r−1} A^{s+1} ⟶^{i} Im^{r} A^{s} ⟶ 0  (0.4)

and

0 ⟶ Im^{r} A^{s+1} ⟶^{⊂} Im^{r−1} A^{s+1} ⟶ \frac{Im j}{B_{r}^{s+r}} ⟶ 0,  (0.5)

where the unmarked arrow is induced by lifting by i^{(r−1)} and applying j. We splice these together (for various r and s), noting that Im j = Ker k, to form the (r−1)-th derived exact couple of (0.1), which consists of the long exact sequences

… Im^{r−1} A^{s−r+2} ⟶^{i} Im^{r−1} A^{s−r+1} ⟶ E_{r}^{s} ⟶^{k} Im^{r−1} A^{s+1} ⟶^{i} Im^{r−1} A^{s} … (0.6)

The differential of this derived exact couple is

d_{r}: E_{r}^{s} ⟶^{k} Im^{r−1} A^{s+1} ⟶ E_{r}^{s+r}.

§1 Limits and colimits 4

It has degree deg(j)+deg(k) in the usual case that deg(i) = 0. We read off the cycles of d_{r} as Ker d_{r} = Z_{r+1}^{s}/B_{r}^{s} and the boundaries as

Im[d_{r}: E_{r}^{s} → E_{r}^{s+r}] = \frac{B_{r+1}^{s+r}}{B_{r}^{s+r}} ≅ \frac{Z_{r}^{s}}{Z_{r+1}^{s}} ≅ \frac{Ker[Im^{r−1} A^{s+1} → A^{s}]}{Ker[Im^{r} A^{s+1} → A^{s}]},  (0.7)

and thus identify the homology of d_{r} at E_{r}^{s} with Z_{r+1}^{s}/B_{r+1}^{s} = E_{r+1}^{s}. We have the spectral sequence r ↦ (E_{r}, d_{r}), defined for r ≥ 1, with the requisite homology isomorphisms H(E_{r}, d_{r}) ≅ E_{r+1}. Our main focus will be on what happens to E_{r} as r ↦ ∞; this is the convergence problem, and is the subject of Part II.

Morphisms of spectral sequences It is clear that a morphism f: A^{s} → \overline{A}^{s} etc. of unrolled exact couples (in the obvious sense) induces a morphism of spectral sequences, which we write as f_{r}: E_{r} → \overline{E}_{r} for r ≥ 1. If f_{k} is an isomorphism (typically for k = 1 or k = 2), so is f_{r} for all r ≥ k. We use this observation frequently.

Part I — TOOLS

In §1, we discuss limits and colimits of abelian groups, and in §4, we do the same for spectra. An important special case is §2 on filtered groups, where the results simplify and become more transparent. We apply this in §3 to the image structure of a sequence of abelian groups, which is far richer than one might expect.

1 Limits and colimits

We review standard material on limits and colimits, especially derived limits. One general reference is Eilenberg–Moore [5]. Our applications usually involve graded groups, although here the grading can safely be ignored.

Given a sequence A of (graded) abelian groups A^{s} and homomorphisms i: A^{s+1} → A^{s} as in diagram (0.2), we have the limit (historically, inverse limit), which we write

A^{∞} = lim_{s} A^{s}.

It comes equipped with homomorphisms ϵ^{s}: A^{∞} → A^{s} that satisfy i^{∘} ϵ^{s+1} = ϵ^{s} and are universal. Explicitly, an element x ∈ A^{∞} may be constructed as a family of elements x^{s} ∈ A^{s} for each s that are compatible in the sense that ix^{s+1} = x^{s} for all s; then ϵ^{s}x = x^{s}. The limit depends only on the portion of diagram (0.2) with s ≥ s_{0}; indeed, A^{s} need only be defined for s ≥ s_{0}.

Dually, we have the colimit (historically, direct limit),

A^{−∞} = colim_{s} A^{s},

together with homomorphisms η^{s}: A^{s} → A^{−∞} that satisfy η^{s} ^{∘} i = η^{s+1} and are universal. Explicitly, every element of A^{−∞} has the form η^{s}a for some s and some a ∈ A^{s}, and η^{s}a = η^{t}b if and only if i^{(s+n)}a = i^{(t+n)}b in A^{−n} for sufficiently large n. The colimit depends only on the portion of diagram (0.2) with s ≤ s_{0}; in fact, A^{s} need only be defined for s ≤ s_{0}.

Exactness A short exact sequence

0 ⟶ A ⟶ B ⟶ C ⟶ 0  (1.1)

in the category of sequences of graded groups is simply one in which

0 ⟶ A^{s} ⟶ B^{s} ⟶ C^{s} ⟶ 0  (1.2)

is a short exact sequence of graded groups for each s. It follows easily from our description that the colimit functor is exact, that

0 ⟶ A^{−∞} ⟶ B^{−∞} ⟶ C^{−∞} ⟶ 0

is a short exact sequence. In contrast, the limit functor is only left exact. Therefore we introduce the derived limit RA^{∞} = Rlim_{s} A^{s} of A; we write it this way (rather than the traditional lim_{s}^{1} A^{s}) to remind that it is the first right-derived functor of lim_{s}. It too depends only on the portion of diagram (0.2) with s ≥ s_{0}. (This fact can be deduced from Theorem 1.4 and Proposition 1.8, below.) Following Milnor [12], both A^{∞} and RA^{∞} are conveniently constructed by means of the exact sequence

0 ⟶ A^{∞} ⟶^{⊂} ∏_{s} A^{s} ⟶^{1−i} ∏_{s} A^{s} ⟶ RA^{∞} ⟶ 0,  (1.3)

where the product ∏_{s} A^{s} is formed degreewise and i denotes the product homomorphism ∏_{s} i: ∏_{s} A^{s} → ∏_{s} A^{s−1}.

Theorem 1.4 Given the short exact sequence (1.1) of graded groups, there is a natural connecting homomorphism δ: C^{∞} → RA^{∞} and a long exact sequence

0 ⟶ A^{∞} ⟶ B^{∞} ⟶ C^{∞} ⟶^{δ} RA^{∞} ⟶ RB^{∞} ⟶ RC^{∞} ⟶ 0. (1.5)

In particular, the functor Rlim is right exact.

Corollary 1.6 In diagram (1.2), assume only that the sequences

A^{s} ⟶ B^{s} ⟶ C^{s} ⟶ 0

are exact. If RA^{∞} = 0, then B^{∞} → C^{∞} is epic.

Proof We apply the Theorem to the short exact sequences 0 → I^{s} → B^{s} → C^{s} → 0 and 0 → K^{s} → A^{s} → I^{s} → 0, where K^{s} = Ker[A^{s} → B^{s}] and I^{s} = Im[A^{s} → B^{s}], to see that RI^{∞} = 0.

The Theorem may be viewed as an application to copies of (1.3) of the following lemma, which is an exercise in diagram chasing. (In applications, it is often necessary to extend a given diagram by zeros.)

Lemma 1.7 Suppose given the commutative diagram with exact rows

[figure: A^{5}; A^{4}; A^{3}; A^{2}; A^{1}; B^{5}; B^{4}; B^{3}; B^{2}; B^{1}; g^{5}; g^{4}; g^{3}; g^{2}; g^{1}]

§2 Filtered groups 6

Put K^{s} = Ker g^{s} and C^{s} = Coker g^{s} for each s; then there is a canonical isomorphism between the homology of K^{3} → K^{2} → K^{1} at K^{2} and the homology of C^{5} → C^{4} → C^{3} at C^{4}.

In particular, K^{3} → K^{2} → K^{1} is exact if and only if C^{5} → C^{4} → C^{3} is exact.

Our policy is never to mention A^{∞} for any sequence A without also introducing RA^{∞}. Usually, one hopes that RA^{∞} = 0 so that (1.5) reduces to a short exact sequence; we need a test for this.

Proposition 1.8 Suppose that i: A^{s+1} → A^{s} is epic for all s ≥ s_{0}. Then: (a) ϵ^{s}: A^{∞} → A^{s} is epic for all s ≥ s_{0};

(b) RA^{∞} = 0.

Proof In (a), we seek x ∈ A^{∞}, with ϵ^{s}x = x^{s} ∈ A^{s} given. We choose x^{s+n} for n > 0 by induction on n to satisfy ix^{s+n} = x^{s+n−1}, and must take x^{s−n} = i^{(n)}x^{s}.

In (b), given y ∈ ∏_{s} A^{s}, we need to solve x − ix = y for x. We start with x^{s} = 0, for some s ≥ s_{0}. By induction on n for n > 0, we choose x^{s+n} to satisfy ix^{s+n} = x^{s+n−1} − y^{s+n−1}, and must take x^{s−n} = y^{s−n} + ix^{s−n+1}.

Remark Thus the vanishing of Coker i for all s implies that Coker ϵ^{s} = 0 for all s and that RA^{∞} = 0. Otherwise, although it is clear that knowledge of the groups Coker i imposes some restriction on what groups Coker ϵ^{s} and RA^{∞} are possible, there appears to be no simple relation.

The following trivial consequence is rather useful.

Corollary 1.9 Suppose that i: A^{s+1} → A^{s} is epic for all s, and that A^{∞} = 0. Then A^{s} = 0 for all s.

In general, limits preserve products. It follows directly from diagram (1.3) that derived limits do too. Given for each λ ∈ Λ a sequence A(λ) of groups A(λ)^{s} and homomorphisms i as in diagram (0.2), we form the product sequence A with groups A^{s} = ∏_{λ} A(λ)^{s} and the evident product homomorphisms i: A^{s+1} → A^{s}.

Proposition 1.10 For the product A of the sequences A(λ) as above, we have lim_{s} A^{s} = ∏_{λ} lim_{s} A(λ)^{s} and Rlim_{s} A^{s} = ∏_{λ} Rlim_{s} A(λ)^{s}.

2 Filtered groups

The derived limit of a general sequence (0.2) is admittedly difficult to interpret. In the special case of a filtration, however, limits, colimits and derived limits all become quite direct and immediately useful. Again, our applications are graded, but the grading may be ignored here.

A decreasing filtration of a (graded) group G consists of subgroups F^{s} = F^{s}G for all integers s, such that F^{s+1} ⊂ F^{s} for all s. There are three desirable properties.

Definition 2.1 The filtration exhausts G (or is exhaustive) if G = ⋃_{s} F^{s}. The filtration is Hausdorff if ⋂_{s} F^{s} = 0. The filtration is complete if every Cauchy sequence in G converges. (We warn that we do not require a complete filtration to be Hausdorff, so the limit of a Cauchy sequence need not be unique.)

To explain the last two concepts, we topologize G by taking the cosets x + F^{s} of F^{s} for all x ∈ G and all s as basic open sets. If ⋂_{s} F^{s} = 0 and x ≠ y, we choose s such that x − y ∉ F^{s}; then x + F^{s} and y + F^{s} are disjoint neighborhoods of x and y, and the space G is Hausdorff. Conversely, if x ≠ 0 and G is Hausdorff, there is a neighborhood 0 + F^{s} of 0 that does not contain x, so x ∉ ⋂_{s} F^{s}. A Cauchy sequence n ↦ x_{n} is one in which x_{m} − x_{n} → 0 as m, n → ∞.

The next result is fundamental and relates all three concepts to the sequence of groups F^{s}. We make heavy use of it, generally without comment.

Proposition 2.2 Suppose we are given a decreasing filtration of G by subgroups F^{s} ⊂ G.

(a) Put F^{−∞} = ⋃_{s} F^{s}; then F^{−∞} = colim_{s} F^{s}, η^{s}: F^{s} → F^{−∞} is the inclusion, and the filtration exhausts G if and only if F^{−∞} = G.

(b) Put F^{∞} = ⋂_{s} F^{s}; then F^{∞} = lim_{s} F^{s}, ϵ^{s}: F^{∞} → F^{s} is the inclusion, and the filtration is Hausdorff if and only if F^{∞} = 0.

(c) Put RF^{∞} = Rlim_{s} F^{s}; then the filtration is complete if and only if RF^{∞} = 0. The limit of a Cauchy sequence is unique if and only if F^{∞} = 0.

Proof Parts (a) and (b) are clear. For (c), take a Cauchy sequence n ↦ x_{n} in G. For each s, x_{m} − x_{n} ∈ F^{s} for all large m and n, so that the image of x_{n} in G/F^{s} is constant for large n and defines an element y^{s} ∈ G/F^{s}. As s varies, we obtain an element y ∈ lim_{s} G/F^{s}, and the question of convergence reduces to lifting y to G. We apply lim_{s} by Theorem 1.4 to the short exact sequence

0 ⟶ F^{s} ⟶ G ⟶ G/F^{s} ⟶ 0

to obtain the long exact sequence

0 ⟶ F^{∞} ⟶ G ⟶ lim_{s} G/F^{s} ⟶ RF^{∞} ⟶ 0,  (2.3)

which yields the whole of (c).

There is much flexibility available in computing F^{∞} and RF^{∞}.

Proposition 2.4 Suppose given a filtered group G and a subgroup K ⊂ F^{∞}. Then for the quotient filtration of G/K by the subgroups F^{s}/K:

(a) colim_{s}(F^{s}/K) = F^{−∞}/K, and exhaustiveness is unaffected;

(b) lim_{s}(F^{s}/K) = F^{∞}/K;

(c) Rlim_{s}(F^{s}/K) = RF^{∞}, and completeness is unaffected.

Proof The exactness of colimits gives (a). For (b) and (c), we apply lim_{s} by Theorem 1.4 and Proposition 1.8 to the short exact sequence

0 ⟶ K ⟶ F^{s} ⟶ F^{s}/K ⟶ 0.

Reconstitution One often wishes to recover the group G from its subquotients F^{t}/F^{s} for t < s. The first two conditions in Definition 2.1 are obviously essential, while the relevance of completeness is clear from diagram (2.3).

Proposition 2.5 Suppose the filtration of G is complete Hausdorff and exhausts G. Then we can reconstruct G from the subquotients F^{t}/F^{s} as

G = lim_{s} G/F^{s} = lim_{s} colim_{t} F^{t}/F^{s}.

Proof The first equality comes from (2.3). The second is Proposition 2.4(a).

We also need a comparison theorem. Suppose given another group \overline{G} filtered by subgroups F^{s}\overline{G} = \overline{F}^{s}, and a filtered homomorphism f: G → \overline{G} (in the strict sense that f(F^{s}) ⊂ \overline{F}^{s} for all s).

Theorem 2.6 Let f: G → \overline{G} be a homomorphism of filtered groups such that:

(i) both filtrations are exhaustive; (ii) f induces F^{∞} ≅ \overline{F}^{∞} (e. g. if both filtrations are Hausdorff ); (iii) G is complete.

Suppose f induces isomorphisms F^{s}/F^{s+1} ≅ \overline{F}^{s}/\overline{F}^{s+1} for all s. Then f is an isomorphism of filtered groups (meaning f: G ≅ \overline{G} and f: F^{s} ≅ \overline{F}^{s} for all s).

Proof For each t, the commutative diagram of short exact sequences

[figure: 0; \frac{F^{s}}{F^{s+1}}; \frac{F^{t}}{F^{s+1}}; \frac{F^{t}}{F^{s}}; 0; 0; \begin{matrix}\overline{F} & s \\ \overline{F} & s+1 \end{matrix}; \begin{matrix}\overline{F} & t \\ \overline{F} & s+1 \end{matrix}; \begin{matrix}\overline{F} & t \\ \overline{F} & s \end{matrix}; 0; ≅]

shows, by induction on s (starting from s = t+1), that f induces F^{t}/F^{s} ≅ \overline{F}^{t}/\overline{F}^{s} for all finite s > t. We take colimits (unions) as t → −∞ to obtain G/F^{s} ≅ \overline{G}/\overline{F}^{s} for all s. Naturality of (2.3) furnishes the commutative diagram with exact rows

[figure: 0; F^{∞}; G; lim_{s} \frac{G}{F^{s}}; RF^{∞}; 0; 0; \overline{F}^{∞}; \overline{G}; lim_{s} \frac{\overline{G}}{\overline{F}^{s}}; R\overline{F}^{∞}; 0; ≅; f; ≅]

Since RF^{∞} = 0, Lemma 1.7 shows that f: G ≅ \overline{G}. Hence f: F^{s} ≅ \overline{F}^{s}.

Remark It was not necessary to assume \overline{G} complete; this is a consequence.

Completion Proposition 2.5 and Theorem 2.6 are the best possible results. If the filtration of G is not exhaustive, we simply consider the filtration of F^{−∞} by the subgroups F^{s}, which is exhaustive. Similarly, if G is not Hausdorff, we replace it by G/F^{∞}, filtered by the subgroups F^{s}/F^{∞}, without affecting the subquotients F^{t}/F^{s}. If G is not complete, diagram (2.3) suggests how to make it complete. The proposition following shows how completeness is essential in Theorem 2.6.

§3 Image subsequences 9

Definition 2.7 Given any filtered group G, we define the completion of G as Ĝ = lim_{s} G/F^{s}. The homomorphism G → Ĝ induced by the projections G → G/F^{s} is called the completion homomorphism.

Proposition 2.8 We can filter Ĝ by the subgroups F̂^{t} = lim_{s} F^{t}/F^{s}. Then: (a) This filtration of Ĝ is complete Hausdorff;

(b) The completion homomorphism G → Ĝ is a filtered homomorphism that induces isomorphisms F^{t}/F^{s} ≅ F̂^{t}/F̂^{s} of subquotient groups, also G/F^{s} ≅ Ĝ/F̂^{s};

(c) We have G/F^{−∞} ≅ Ĝ/ ⋃_{t} F̂^{t}; in particular, the subgroups F̂^{t} exhaust Ĝ if and only if the subgroups F^{t} exhaust G.

Proof If we apply lim_{s} by Theorem 1.4 to the short exact sequence

0 ⟶ F^{t}/F^{s} ⟶ G/F^{s} ⟶ G/F^{t} ⟶ 0,

the resulting long exact sequence (1.5) simplifies to

0 ⟶ F̂^{t} ⟶ Ĝ ⟶ G/F^{t} ⟶ 0,  (2.9)

because Rlim_{s} F^{t}/F^{s} = 0 by Proposition 1.8(b). So we may regard F̂^{t} as a subgroup of Ĝ, and it has the correct quotient. The rest of (b) follows easily.

Next, we apply lim_{t} to (2.9) to obtain the exact sequence

0 ⟶ lim_{t} F̂^{t} ⟶ Ĝ ⟶ lim_{t} G/F^{t} ⟶ Rlim_{t} F̂^{t} ⟶ 0.

By definition, Ĝ = lim_{t} G/F^{t}, and we have (a). If instead we apply the exact functor colim_{t} to (2.9), we obtain the short exact sequence

0 ⟶ ⋃_{t} F̂^{t} ⟶ Ĝ ⟶ G/F^{−∞} ⟶ 0,

which gives (c).

3 Image subsequences

Given a sequence A of (graded) groups A^{s} as in (0.2), it is important to know whether RA^{∞} = 0. By Proposition 1.8, this is true if each i: A^{s+1} → A^{s} is epic.

We already introduced in equation (0.3) the filtration of A^{s} by the image subgroups Im^{r} A^{s} = Im[i^{(r)}: A^{s+r} → A^{s}] (for r ≥ 0). The more general Mittag–Leffler condition, that Im^{r} A^{s} is independent of r for r ≥ r_{0}(s), also ensures that RA^{∞} = 0.

We generalize further, by asking whether this filtration is complete or Hausdorff.

Definition 3.1 Given the sequence of groups A^{s} as in diagram (0.2), we define sequences Q and RQ by Q^{s} = ⋂_{r} Im^{r} A^{s} = lim_{r} Im^{r} A^{s} and RQ^{s} = Rlim_{r} Im^{r} A^{s}.

This is only the beginning of a very rich structure.

Definition 3.2 Given any sequence A as in (0.2), we define the first image subsequence Im A of A by (Im A)^{s} = Im^{1} A^{s}, as above, with i: (Im A)^{s} → (Im A)^{s−1} defined by restriction. Then for any integer r ≥ 0, we iterate by defining Im^{r+1} A = Im(Im^{r} A), with the result that (Im^{r} A)^{s} = Im^{r} A^{s} as in equation (0.3).

We extend the definition to any ordinal α by transfinite induction,

Im^{α} A = \{ \begin{matrix}A & if α = 0; \\ Im(Im^{β} A) & if α = β + 1; \\ ⋂_{β<α} Im^{β} A & if α is a limit ordinal; \end{matrix}

with (Im^{α} A)^{s+1} → (Im^{α} A)^{s} defined in all cases by restriction from i: A^{s+1} → A^{s}.

We define the image order of A to be the smallest ordinal σ such that Im^{σ+1} A = Im^{σ} A. (It exists because Im^{α} A is a decreasing function of α.)

Then Im^{α} A = Im^{σ} A for all α ≥ σ. From now on, we write (Im^{α} A)^{s} simply as Im^{α} A^{s}; in particular, Im^{ω} A^{s} = Q^{s}. By Proposition 1.8,

Im^{σ} A^{s} = Im[ϵ^{s}: A^{∞} → A^{s}].  (3.3)

These definitions are not vacuous; sequences really can be this complicated, and the phenomenon is not new.

Example The image order σ is arbitrary. Given any ordinal σ, for each s ≥ 0 let A^{s} be free abelian on the set of all (s+1)-tuples (α_{s}, α_{s−1}, …, α_{1}, α_{0}), where the α’s are any ordinals that satisfy 0 ≤ α_{s} < α_{s−1} < … < α_{1} < α_{0} < σ, and take A^{s} = 0 for s < 0. We define i: A^{s} → A^{s−1} (for s > 0) on the generators by omitting the first entry,

i(α_{s}, α_{s−1}, …, α_{1}, α_{0}) = (α_{s−1}, … α_{1}, α_{0}).

By transfinite induction, Im^{α} A^{s} is generated by all (s+1)-tuples that satisfy α_{s} ≥ α. Thus Im^{σ} A = 0, but Im^{α} A ≠ 0 for all α < σ.

Example Given a p-primary torsion abelian group G, where p is a prime, take A^{s} = G for all s and i: A^{s+1} → A^{s} as multiplication by p. Here, Im^{α} A^{s} is written p^{α}G, and the quotient p^{ωα}G/p^{ω(α+1)}G is known as the α-th Ulm factor of G (e. g. Kurosh [7, §27–28]). Then p^{σ}G is the largest divisible subgroup of G.

The Mittag–Leffler exact sequence It is clear by direct construction that lim_{s} Im^{α} A^{s} is independent of α. This is false for Rlim_{s}, however, as Rlim_{s} Im^{σ} A^{s} = 0 by Proposition 1.8. The group Rlim_{s} Im^{α} A^{s} can change at limit ordinals α; we study only the first case, α = ω. (We refrain from writing RQ^{∞} to avoid ambiguity.)

Theorem 3.4 For any sequence (0.2) of groups A^{s} and homomorphisms i: (a) lim_{s} Q^{s} = lim_{s} A^{s} = A^{∞};

(b) We have the Mittag–Leffler short exact sequence

0 ⟶ Rlim_{s} Q^{s} ⟶ RA^{∞} ⟶ lim_{s} RQ^{s} ⟶ 0;  (3.5)

(c) Rlim_{s} RQ^{s} = 0; further, RQ^{s+1} → RQ^{s} is epic for all s.

Corollary 3.6 If RA^{∞} = 0, then RQ^{s} = 0 for all s.

Proof We apply Corollary 1.9 to RQ^{s}.

Proof of Theorem The result can be considered an application of the spectral sequence of the double limit system Im^{r} A^{s}. We offer an elementary version of this proof. As already noted, (a) is clear.

First, we assume the filtration of each A^{s} by the subgroups Im^{r} A^{s} is complete Hausdorff, so that Q^{s} = RQ^{s} = 0. We have to show that RA^{∞} = 0.

We introduce the double sequence of image groups

I^{s,t} = Im[A^{max(s,t)} → A^{s}] = \{ \begin{matrix}Im^{t−s} A^{s} = Im[i^{(t−s)}: A^{t} → A^{s}] & if t ≥ s; \\ A^{s} & if t ≤ s; \end{matrix}

equipped with obvious homomorphisms I^{s,t} → I^{u,v} whenever s ≥ u and t ≥ v. For fixed t, we have I^{s,t} = A^{s} for all large s, so that lim_{s} I^{s,t} = A^{∞} = 0 and Rlim_{s} I^{s,t} = RA^{∞}. Thus the defining exact sequence (1.3) reduces to

0 ⟶ P^{t} ⟶^{1−i} P^{t} ⟶ RA^{∞} ⟶ 0,  (3.7)

where we write P^{t} = ∏_{s} I^{s,t}.

On the other hand, if we fix s, I^{s,t} = Im^{t−s} A^{s} for large t, so that by hypothesis, lim_{t} I^{s,t} = Rlim_{t} I^{s,t} = 0. Then lim_{t} P^{t} = Rlim_{t} P^{t} = 0 by Proposition 1.10. We apply lim_{t} by Theorem 1.4 to (3.7) to deduce that RA^{∞} = 0.

Second, we consider the general case. For each s, we take the completion Â^{s} of the group A^{s}, filtered by the subgroups Im^{r} A^{s}; the exact sequence (2.3) becomes

0 ⟶ Q^{s} ⟶ A^{s} ⟶ Â^{s} ⟶ RQ^{s} ⟶ 0.  (3.8)

By Proposition 2.8(a), Â^{s} has a complete Hausdorff filtration by subgroups F^{r}Â^{s}.

As s varies, the groups Â^{s} form a sequence Â. To identify Im^{r} Â^{s} with F^{r}Â^{s}, we apply lim_{t} by Corollary 1.6 to the exact sequence

Ker[i^{(r)}: A^{s+r} → A^{s}] ⟶ \frac{A^{s+r}}{Im^{t} A^{s+r}} ⟶ \frac{Im^{r} A^{s}}{Im^{r+t} A^{s}} ⟶ 0,

to deduce that Â^{s+r} → F^{r}Â^{s} is epic. Then the first case applies to Â, to show that lim_{s} Â_{s} = Rlim_{s} Â^{s} = 0.

In order to apply Theorem 1.4, we break up (3.8) into two short exact sequences by introducing J^{s} = A^{s}/Q^{s}. The short exact sequence

0 ⟶ J^{s} ⟶ Â^{s} ⟶ RQ^{s} ⟶ 0

yields J^{∞} = 0, RJ^{∞} ≅ lim_{s} RQ^{s}, and Rlim_{s} RQ^{s} = 0. The other short exact sequence,

0 ⟶ Q^{s} ⟶ A^{s} ⟶ J^{s} ⟶ 0,

then immediately yields (b).

To finish (c), we apply the right exact functor Rlim_{r} to the epimorphisms Im^{r} A^{s+1} → Im^{r+1} A^{s} to see that RQ^{s+1} → RQ^{s} is epic.

Remark For higher limit ordinals, the situation is far more complicated. The homological dimension of the functor Rlim_{β}, where β runs over all ordinals less than α, may be arbitrarily large or even infinite.

§4 Homotopy limits and colimits of spectra 12

4 Homotopy limits and colimits of spectra

We recall some standard material on limits and colimits of sequences of spaces and spectra. Much of this is presented in vastly greater generality in e. g. Vogt [14]. We also list some related results on localization and completion of spectra at a set of primes, following Bousfield [3].

Milnor’s results Given a sequence of inclusions i_{s}: X_{s} ⊂ X_{s+1} of spaces or spectra, we can construct their union X_{∞}, with inclusions η_{s}: X_{s} ⊂ X_{∞}. This serves as the colimit colim_{s} X_{s}, before taking homotopy classes.

Any spectrum M defines a generalized homology theory M_{∗}(−). (By M_{∗}(X), we mean the whole graded group, not a generic component of it.) Milnor [12] related M_{∗}(X_{∞}) to the groups M_{∗}(X_{s}) by treating the mapping telescope of the maps i_{s} as a pushout, to construct in effect the exact triangle of spectra

⋁_{s} X_{s} ⟶^{1−i} ⋁_{s} X_{s} ⟶^{η} X_{∞} ⟶^{δ} ⋁_{s} X_{s},  (4.1)

where η|X_{s} = η_{s} for all s, and i denotes the map

i: ⋁_{s≥0} X_{s} \underrightarrow{⋁_{s} i_{s}} ⋁_{s≥0} X_{s+1} = ⋁_{s≥1} X_{s} ⊂ ⋁_{s≥0} X_{s}.

Theorem 4.2 (Milnor) Let X be a space or spectrum, with an increasing filtration by subspaces or subspectra X_{s} that exhaust X, and let M be any spectrum. Then in homology:

(a) The maps η_{s} induce an isomorphism colim_{s} M_{n}(X_{s}) = M_{n}(X);

(b) colim_{s} M_{n}(X, X_{s}) = 0.

Proof For (a), we apply M_{∗}(−) to the exact triangle (4.1). Since 1 − i_{∗} is monic, we obtain the short exact sequence

0 ⟶ ⨁_{s} M_{n}(X_{s}) ⟶^{1−i_{∗}} ⨁_{s} M_{n}(X_{s}) ⟶^{η_{∗}} M_{n}(X) ⟶ 0,

which identifies M_{n}(X) with colim_{s} M_{n}(X_{s}). Part (b) is equivalent, by applying the exact functor colim_{s} to the homology exact sequence of the pair (X, X_{s}).

If we apply M-cohomology M^{∗}(−) instead, we obtain the long exact sequence

… ∏_{s} M^{n−1}(X_{s}) ⟶^{1−i^{∗}} ∏_{s} M^{n−1}(X_{s}) ⟶^{δ^{∗}} M^{n}(X_{∞}) ⟶^{η^{∗}} ∏_{s} M^{n}(X_{s}) …

Comparison with diagram (1.3) immediately yields part (a) of the following theorem, with X = X_{∞}.

Theorem 4.3 (Milnor) Let X be a space or spectrum, with an increasing filtration by subspaces or subspectra X_{s} that exhaust X, and let M be any spectrum. Then in cohomology:

(a) We have the Milnor short exact sequence

0 ⟶ Rlim_{s} M^{n−1}(X_{s}) ⟶^{δ^{∗}} M^{n}(X) ⟶^{η^{∗}} lim_{s} M^{n}(X_{s}) ⟶ 0;  (4.4)

(b) lim_{s} M^{n}(X, X_{s}) = 0 and Rlim_{s} M^{n}(X, X_{s}) = 0.

Although (b) is the obvious analogue of Theorem 4.2(b), it is unclear whether it is equivalent to (a). It is easy to show that (b) implies (a), by breaking up the cohomology exact sequence of the pair (X, X_{s}) into short exact sequences and applying Theorem 1.4 to each. The difficulty with the converse is that δ^{∗} is far from obvious. We defer the proof of (b) until after Proposition 4.5.

Homotopy colimits The presence of the Rlim term in diagram (4.4) shows that in general, X_{∞} is not the colimit of the X_{s} in the stable homotopy category. The diagram does show, however, that X_{∞} has the weak universal property: given compatible maps (cohomology classes) f_{s}: X_{s} → M for each s, there exists a compatible map f_{∞}: X_{∞} → M, but it need not be unique. Since this is the closest one can expect to come to a true colimit, one calls X_{∞} the homotopy colimit and writes hocolim_{s} X_{s}.

We need to be more general. Given any maps i_{s}: X_{s} → X_{s+1} (that preserve skeletons) for s ≥ 0, not necessarily inclusions, we take the mapping telescope itself as X_{∞}, together with inclusions η_{s}: X_{s} ⊂ X_{∞}, and obtain the same exact triangle (4.1). We may use the triangle to define the homotopy colimit hocolim_{s} X_{s} as X_{∞}, uniquely up to homotopy, and parts (a) of Theorems 4.2 and 4.3 remain valid.

Also, for each s, we form the mapping telescope T_{s} of

X_{s} ⟶^{i_{s}} X_{s+1} ⟶^{i_{s+1}} X_{s+2} ⟶ …,

which has η_{s}: X_{s} ⊂ T_{s}. By naturality, the commutative diagram

[figure: X_{s}; X_{s+1}; X_{s+2}; …; X_{s+1}; X_{s+2}; X_{s+3}; …; i_{s}; i_{s}; i_{s+1}; i_{s+1}; i_{s+2}; i_{s+1}; i_{s+2}]

induces a map g_{s}: (T_{s}, X_{s}) → (T_{s+1}, X_{s+1}) of mapping telescopes, and hence a map Y_{s} → Y_{s+1}, where Y_{s} = T_{s}/X_{s}.

Proposition 4.5 Given maps i_{s}: X_{s} → X_{s+1} as above, we construct the mapping telescopes T_{s}, the spectra Y_{s} = T_{s}/X_{s}, and maps g_{s}: T_{s} → T_{s+1}. Then:

(a) Each g_{s}: T_{s} → T_{s+1} is a homotopy equivalence;

(b) The homotopy colimit hocolim_{s} Y_{s} is trivial;

(c) For any spectrum M, lim_{s} M^{∗}(Y_{s}) = 0 and Rlim_{s} M^{∗}(Y_{s}) = 0.

Proof In (a), the obvious inclusion T_{s+1} ⊂ T_{s} is a homotopy inverse to g_{s}. Form the mapping telescope T_{∞} of the maps g_{s}, so that X_{∞} ⊂ T_{∞}. Then the mapping telescope of the spectra Y_{s} is T_{∞}/X_{∞}. There is a retraction r: T_{∞} → X_{∞} that satisfies r|T_{0} = 1 (recall that T_{0} = X_{∞}). From (a), η_{0}: T_{0} ⊂ T_{∞} is a homotopy equivalence; it follows that X_{∞} ⊂ T_{∞} is also a homotopy equivalence, and we have (b). For (c), we apply Theorem 4.3(a) to Y_{s}.

From (a), we see that Y_{s} may be defined up to homotopy by the exact triangle

X_{s} ⟶^{η_{s}} X_{∞} ⟶ Y_{s} ⟶ X_{s}.  (4.6)

Hence M^{∗}(Y_{s}) ≅ M^{∗}(X_{∞}, X_{s}), and Theorem 4.3(b) follows.

Homotopy limits Dually, given any sequence of maps

… ⟶ X^{3} ⟶^{i^{3}} X^{2} ⟶^{i^{2}} X^{1} ⟶^{i^{1}} X^{0}

of spectra, we define the homotopy limit spectrum X^{∞} = holim_{s} X^{s}, uniquely up to homotopy, by the exact triangle

∏_{s} X^{s} ⟶ X^{∞} ⟶^{ϵ} ∏_{s} X^{s} ⟶^{1−i} ∏_{s} X^{s},  (4.7)

in which we arrange deg(ϵ) = 0. We often wish to cancel out X^{∞}. Dually to (4.6), we define spectra Y^{s}, uniquely up to homotopy, by the exact triangles

X^{∞} ⟶^{ϵ^{s}} X^{s} ⟶^{f^{s}} Y^{s} ⟶ X^{∞}.  (4.8)

We fill in maps j^{s+1}: Y^{s+1} → Y^{s} to form morphisms between these exact triangles, and dualize Theorem 4.3.

Theorem 4.9 Suppose given any maps i^{s}: X^{s} → X^{s−1}. Then:

(a) For any spectrum W, we have the Milnor-type short exact sequence

0 ⟶ Rlim_{s}{W, X^{s}}^{n−1} ⟶ {W, X^{∞}}^{n} ⟶^{ϵ_{∗}} lim_{s}{W, X^{s}}^{n} ⟶ 0.  (4.10)

(b) For a good choice of the maps j^{s+1}: Y^{s+1} → Y^{s} we have holim_{s} Y^{s} = 0, and hence lim_{s}{W, Y^{s}}^{n} = 0 and Rlim_{s}{W, Y^{s}}^{n} = 0 for all W.

Proof For (a), we map W into the exact triangle (4.7). For (b), we dualize Proposition 4.5, by turning inclusions (cofibrations), wedges and pushouts into fibrations, products and pullbacks. (To make this work, we choose some version of the stable category that has the necessary structure, such as any of the modern model category candidates. Even Kan’s category of simplicial spectra [6] will do.) Note that all maps and homotopies in Proposition 4.5 are canonical and natural.

Localization and completion of spectra We recall some standard tools from Bousfield [3, §2]. For any abelian group G, denote by L(G) the associated Moore spectrum, which is (−1)-connected and has H_{0}(L(G)) = G as its only nonzero homology group.

Given any set J of primes, denote by ℤ_{J} the ring of all rational numbers a/b ∈ ℚ such that b has no prime factor in J. (If J is the set of all primes, ℤ_{J} reduces to the integers ℤ.) The J-localization of any spectrum Y is simply Y ∧ L(ℤ_{J}), and is equipped with the localization map Y = Y ∧ S → Y ∧ L(ℤ_{J}) = Y_{J}, where S denotes the sphere spectrum. This map induces an isomorphism {Y_{J}, Z}^{∗} ≅ {Y, Z}^{∗} for any J-local spectrum Z = W_{J}. The most important case is the p-localization Y_{(p)} of Y, when J consists of the single prime p and ℤ_{J} is written ℤ_{(p)}. Two other important cases are Y [p^{−1}], the localization away from p, when J consists of all primes except p and ℤ_{J} = ℤ[p^{−1}], and the rationalization Y_{∅} of Y, when ℤ_{J} = ℚ.

We also need the p-completion Ŷ_{(p)} of Y. In the common case when π_{n}(Y) is finitely generated for all n, this is found to be Y ∧ Ŝ_{(p)}, where Ŝ_{(p)} denotes the Moore spectrum L(ℤ̂_{(p)}) for the p-adic integers ℤ̂_{(p)}; for general Y, it is the function spectrum F (Σ^{−1}L(ℤ/p^{∞}), Y), where ℤ/p^{∞} = ℤ[p^{−1}]/ℤ. It is equipped with the completion map Y → Ŷ_{(p)}, which factors through the p-localization Y_{(p)} because Ŷ_{(p)} is clearly p-local. It induces an isomorphism {Ŷ_{(p)}, Z}^{∗} ≅ {Y, Z}^{∗} for any p-complete spectrum Z = Ŵ_{(p)}. For a set J of primes, the J-completion Ŷ_{J} of Y is simply the product ∏_{p∈J} Ŷ_{(p)}.

There are significant cases where completion and localization are not needed.

Proposition 4.11 Assume that π_{n}(X) is a p-primary torsion group for all n (e. g. if X is connective and H_{n}(X) is a p-primary torsion group for all n). Then the localization and completion maps induce isomorphisms

{X, Y }^{∗} ≅ {X, Y_{(p)}}^{∗} ≅ {X, Ŷ_{(p)}}^{∗}.

Proof The arithmetic square of localization maps

[figure: Y; Y_{(p)}; Y [p^{−1}]; Y_{∅}]

induces the Mayer–Vietoris-type exact sequence

… {X, Y_{∅}}^{∗} ⟶ {X, Y }^{∗} ⟶ {X, Y_{(p)}}^{∗} ⊕ {X, Y [p^{−1}]}^{∗} ⟶ {X, Y_{∅}}^{∗} …

Since X[p^{−1}] is trivial, {X, Y [p^{−1}]}^{∗} ≅ {X[p^{−1}], Y [p^{−1}]}^{∗} = 0. Similarly, we find {X, Y_{∅}}^{∗} = 0, and the exact sequence simplifies to the first isomorphism. For the second, according to Bousfield [3, Prop. 2.5], we need to show that {X, F (L(ℤ[p^{−1}]), Y)}^{∗} vanishes. By definition,

{X, F (L(ℤ[p^{−1}]), Y)}^{∗} = {X ∧ L(ℤ[p^{−1}]), Y }^{∗} = {X[p^{−1}], Y }^{∗} = 0.

Part II — CONVERGENCE

In this Part, we study the convergence of the spectral sequence arising from the unrolled exact couple (0.1). In §5, we define the relevant groups and develop their properties.

§5 Types of convergence 16

The generality of §5 is irrelevant to the vast majority of real-world spectral sequences. It also lacks transparency. We therefore show how the results simplify in the situations that occur most frequently. In one common graphical presentation, the component E_{r}^{s,t} of E_{r}^{s} in degree s + t is placed at the point (s, t) in the plane. Many spectral sequences occupy only a quadrant or at most a half-plane. In each case, we have a convergence theorem and a comparison theorem for morphisms of spectral sequences. Some examples are given in Part III.

In §6, we discuss half-plane spectral sequences with exiting differentials. These are the easiest to handle because there are no problems with limits or derived limits; we have nothing to add and include this section merely for completeness.

In §7, we discuss half-plane spectral sequences with entering differentials; this is the case where our results are of most interest.

In §8, we attack the general case of whole-plane spectral sequences. For many years, they were considered so intractable as to be virtually useless. (The material in the preprint editions on their convergence was highly speculative, with no application in sight.) This attitude is changing, as significant examples now exist that are not amenable to elementary treatment, and more can be expected.

5 Types of convergence

Suppose given any spectral sequence r ↦ (E_{r}, d_{r}), defined for r ≥ r_{0}. In order to discuss its convergence, we write E_{r}^{s} = Z_{r}^{s}/B_{r}^{s}, where 0 ⊂ B_{r}^{s} ⊂ Z_{r}^{s} ⊂ E_{r_{0}}^{s}, and introduce the (graded) groups:

Z_{∞}^{s} = ⋂_{r} Z_{r}^{s} = lim_{r} Z_{r}^{s}, the group of infinite cycles;
B_{∞}^{s} = ⋃_{r} B_{r}^{s} = colim_{r} B_{r}^{s}, the group of infinite boundaries;
E_{∞}^{s} = Z_{∞}^{s}/B_{∞}^{s} ≅ (Z_{∞}^{s}/B_{m}^{s})/(B_{∞}^{s}/B_{m}^{s}), which form the E_{∞}-term;
RE_{∞}^{s} = Rlim_{r} Z_{r}^{s} ≅ Rlim_{r}(Z_{r}^{s}/B_{m}^{s}), which form the derived E_{∞}-term.

(5.1)

The term RE_{∞} is suggested by our policy of introducing the associated derived limit along with every limit. The second versions of E_{∞} and RE_{∞} (provided by Proposition 2.4, for any m ≥ r_{0}) show that they depend only on the terms E_{r} of the spectral sequence with r ≥ m.

Typically, one relates the term E_{∞} to some target graded group G, equipped with a decreasing filtration by subgroups F^{s} = F^{s}G, by comparing E_{∞}^{s} with F^{s}/F^{s+1}. To indicate that the target group being considered is G, we write

E_{r_{0}}^{s} ⟹ G,

without implying that the target is in any sense hit; any information on convergence is to be stated separately. We have the groups F^{−∞}, F^{∞} and RF^{∞}, which Proposition 2.2 interprets.

We adopt the terminology of Cartan–Eilenberg [4, Chap. XV, §2].

Definition 5.2 Given a spectral sequence r ↦ (E_{r}, d_{r}) and a filtered target group G, we say the spectral sequence:

(i) converges weakly to G if the filtration exhausts G (i. e. F^{−∞} = G) and we have isomorphisms E_{∞}^{s} ≅ F^{s}/F^{s+1} for all s (possibly with nonzero degree); (ii) converges to G if (i) holds and the filtration of G is Hausdorff (i. e. F^{∞} = 0); (iii) converges strongly to G if (i) holds and the filtration of G is complete Haus-

dorff (i. e. F^{∞} = RF^{∞} = 0).

Strong convergence allows us to reconstruct G from the term E_{∞}, up to group extension, by means of Proposition 2.5. It is even more useful in comparing spectral sequences, in view of Theorem 2.6.

Theorem 5.3 Suppose given a morphism f of spectral sequences, with components f_{r}: (E_{r}, d_{r}) → (\overline{E}_{r}, \overline{d}_{r}), where (E_{r}, d_{r}) converges strongly to G and (\overline{E}_{r}, \overline{d}_{r}) converges to \overline{G} (not necessarily strongly), together with a compatible morphism f: G → \overline{G} of filtered target groups. If f_{m}: E_{m} → \overline{E}_{m} is an isomorphism for some m ≤ ∞, then f: G → \overline{G} is an isomorphism of filtered groups.

Two filtered groups To say more, we assume from now on that our spectral sequence comes from the unrolled exact couple (0.1). There are two clear candidates for the target group, A^{−∞} and A^{∞}, each with an obvious filtration. Both are useful. (And if we apply an exact contravariant functor to the unrolled exact couple to obtain another one, A^{∞} and A^{−∞} switch places, if we are lucky.) One generally tries to arrange for one of these groups to be zero; for if we replace each A^{s} by A^{s} ⊕ K, the new unrolled exact couple has the same spectral sequence, but has A^{±∞} replaced by A^{±∞} ⊕ K.

Lemma 5.4 Take any unrolled exact couple as in diagram (0.1). Then:

(a) The filtration of the colimit A^{−∞} = colim_{s} A^{s} by the subgroups F^{s}A^{−∞} = Im[η^{s}: A^{s} → A^{−∞}] exhausts A^{−∞} (but need not be complete or Hausdorff);

(b) The filtration of the limit A^{∞} = lim_{s} A^{s} by the subgroups F^{s}A^{∞} = Ker[ϵ^{s}: A^{∞} → A^{s}] is complete Hausdorff, and F^{−∞}A^{∞} = Ker[A^{∞} → A^{−∞}] (so that this filtration need not exhaust A^{∞}, but does if A^{−∞} = 0).

Proof Part (a) is immediate from our description of A^{−∞} in §1.

For (b), we need to show that lim_{s} F^{s}A^{∞} = Rlim_{s} F^{s}A^{∞} = 0. We apply lim_{s} by Theorem 1.4 to the short exact sequence

0 ⟶ F^{s}A^{∞} ⟶ A^{∞} ⟶ Im^{σ} A^{s} ⟶ 0,  (5.5)

which comes from equation (3.3), to obtain the exact sequence

0 ⟶ lim_{s} F^{s}A^{∞} ⟶ A^{∞} ⟶ lim_{s} Im^{σ} A^{s} ⟶ Rlim_{s} F^{s}A^{∞} ⟶ 0.

But lim_{s} Im^{σ} A^{s} = A^{∞}. Finally, exactness of colimits gives

F^{−∞}A^{∞} = colim_{s} Ker[ϵ^{s}: A^{∞} → A^{s}] = Ker[A^{∞} → A^{−∞}].

Both filtrations are closely related to the E_{∞}-term of the spectral sequence. We recall from Definition 3.1 the groups Q^{s} = Im^{ω} A^{s} and RQ^{s} = Rlim_{r} Im^{r} A^{s}.

Lemma 5.6 We have the exact sequences

0 ⟶ \frac{F^{s}A^{−∞}}{F^{s+1}A^{−∞}} ⟶ E_{∞}^{s} ⟶ Q^{s+1} ⟶^{i} Q^{s} ⟶ RE_{∞}^{s} ⟶ RQ^{s+1} ⟶ RQ^{s} ⟶ 0

(5.7) and

0 ⟶ \frac{F^{s}A^{∞}}{F^{s+1}A^{∞}} ⟶ Im^{σ} A^{s+1} ⟶^{i} Im^{σ} A^{s} ⟶ 0.  (5.8)

Proof We apply lim_{r} by Theorem 1.4 to the short exact sequence (0.4) to obtain (with the help of Proposition 2.4) the exact sequence

0 ⟶ \frac{Z_{∞}^{s}}{Ker k} ⟶ Q^{s+1} ⟶^{i} Q^{s} ⟶ RE_{∞}^{s} ⟶ RQ^{s+1} ⟶ RQ^{s} ⟶ 0.

We splice this with the short exact sequence

0 ⟶ \frac{Im j}{B_{∞}^{s}} ⟶ \frac{Z_{∞}^{s}}{B_{∞}^{s}} ⟶ \frac{Z_{∞}^{s}}{Ker k} ⟶ 0,

in which Im j = Ker k and Z_{∞}^{s}/B_{∞}^{s} = E_{∞}^{s}. We recall that B_{∞}^{s} = j Ker[η^{s}: A^{s} → A^{−∞}], so that j: A^{s} → E^{s} and η^{s}: A^{s} → A^{−∞} induce isomorphisms

\frac{Im j}{B_{∞}^{s}} ⟵^{≅} \frac{A^{s}}{Im i + Ker[η^{s}: A^{s} → A^{−∞}]} ⟶^{≅} \frac{F^{s}A^{−∞}}{F^{s+1}A^{−∞}} .

The short exact sequence (5.8) follows immediately from (5.5).

If RE_{∞} = 0, there are significant simplifications.

Lemma 5.9 Suppose that the spectral sequence defined by the unrolled exact couple (0.1) satisfies RE_{∞} = 0. Then:

(a) ϵ^{s}: A^{∞} → Q^{s} is epic for all s and Im^{σ} A^{s} = Q^{s} (so that σ ≤ ω).

(b) The natural homomorphism RA^{∞} → RQ^{s} is an isomorphism for all s.

(c) The following conditions are equivalent: (i) The spectral sequence converges weakly to A^{−∞}; (ii) ϵ^{s}: A^{∞} → A^{s} is monic for all s; (iii) ϵ^{s} induces an isomorphism A^{∞} ≅ Q^{s} for all s.

Proof The exact sequence (5.7) breaks up into the exact sequence

0 ⟶ \frac{F^{s}A^{−∞}}{F^{s+1}A^{−∞}} ⟶ E_{∞}^{s} ⟶^{k} Q^{s+1} ⟶^{i} Q^{s} ⟶ 0

and the isomorphism RQ^{s+1} ≅ RQ^{s}. Since A^{∞} = lim_{s} Q^{s}, Proposition 1.8 and equation (3.3) give (a) and the Mittag–Leffler short exact sequence (3.5) simplifies to (b). Then in (c), (ii) and (iii) are equivalent by (a), and the above exact sequence shows that (i) and (iii) are equivalent.

Conditional convergence There are two variants of our main definition. As already said, it is highly desirable to arrange A^{∞} = 0 or A^{−∞} = 0. Our policy on limits suggests strengthening the first condition.

§6 Half-plane spectral sequences with exiting differentials 19

Definition 5.10 Given the unrolled exact couple (0.1), we say that the resulting spectral sequence converges conditionally to the colimit A^{−∞} if A^{∞} = 0 and RA^{∞} = 0. We say the spectral sequence converges conditionally to the limit A^{∞} if A^{−∞} = 0.

The convergence is conditional in the sense that, in the presence of extra hypotheses that are often easily verified in practice, Theorems 7.1 and 8.2 deliver strong convergence, with all its advantages. Easy examples show that by itself, conditional convergence does not even guarantee weak convergence. We shall find that it holds in many important applications for general structural reasons, not for finiteness or computational reasons. Moreover, we have results for conditionally convergent spectral sequences that do not require or imply any of the usual forms of convergence.

In this situation too, Lemma 5.6 simplifies usefully.

Lemma 5.11 Suppose the spectral sequence resulting from the unrolled exact couple (0.1) converges conditionally to the colimit A^{−∞}. Then:

(a) RQ^{s} = 0 for all s;

(b) The filtration of A^{−∞} is complete (but need not be Hausdorff);

(c) We have the long exact sequence

0 ⟶ \frac{F^{s}A^{−∞}}{F^{s+1}A^{−∞}} ⟶ E_{∞}^{s} ⟶ Q^{s+1} ⟶^{i} Q^{s} ⟶ RE_{∞}^{s} ⟶ 0.  (5.12)

Proof Since RA^{∞} = 0, RQ^{s} = 0 for all s by Corollary 3.6, and diagram (5.7) shortens as in (c).

For (b), we apply the right exact functor Rlim_{s} to the epimorphism η^{s}: A^{s} → F^{s}A^{−∞} to obtain the epimorphism 0 = RA^{∞} → RF^{∞}. Thus RF^{∞} = 0.

6 Half-plane spectral sequences with exiting differentials

We define these to be spectral sequences that occupy a half-plane in such a way that all except finitely many of the differentials leaving any point (s, t) of the halfplane exit the half-plane, and so automatically vanish. We trivially have RE_{∞} = 0. (They were previously called left half-plane spectral sequences, but are often reindexed for convenience to shift them into some other half-plane.)

Typically, they arise from an unrolled exact couple (0.1) in which E^{s} = 0 for all s > 0; all the derived limit groups vanish automatically, including RA^{∞}. This is the only situation we discuss in detail. (However, other exact couples can produce them. The results generalize appropriately, as all arguments can be carried out degreewise; the main difficulty is to find notation that would help rather than hinder the exposition.)

Theorem 6.1 Suppose the unrolled exact couple (0.1) satisfies E^{s} = 0 for all s > 0 (equivalently, ϵ^{s}: A^{∞} ≅ A^{s} for all s > 0), and so yields a half-plane spectral sequence with exiting differentials.

(a) If A^{∞} = 0, the spectral sequence converges strongly to the colimit G = A^{−∞}, with isomorphisms F^{s}G/F^{s+1}G ≅ E_{∞}^{s};

(b) If A^{−∞} = 0, the spectral sequence converges strongly to the limit G = A^{∞}, with isomorphisms F^{s}G/F^{s+1}G ≅ E_{∞}^{s}.

§7 Half-plane spectral sequences with entering differentials 20

Proof In either case, the filtration of G is trivially complete Hausdorff, since F^{1}G = 0, and exhausts G by Lemma 5.4.

In (a), A^{s} = 0 for all s > 0, hence Q^{s} = 0 for all s. Then diagram (5.7) reduces to the desired isomorphism.

In (b), we have

Im^{r} A^{s} = Im[i^{(r)}: A^{s+r} → A^{s}] = \{ \begin{matrix}A^{s} & if s ≥ 1; \\ Im^{1−s} A^{s} & if s ≤ 1 and r ≥ 1−s. \end{matrix}

Thus Im^{σ} A^{s} = Q^{s} for all s. Now we compare diagrams (5.8) and (5.7).

No new comparison theorem is needed for such spectral sequences, because Theorem 5.3 is entirely satisfactory.

7 Half-plane spectral sequences with entering differentials

We define these to be spectral sequences that occupy a half-plane in such a way that all except finitely many of the differentials that arrive at any point (s, t) of the half-plane originate at points outside the half-plane, and so automatically vanish. (They were previously called right half-plane spectral sequences.)

Typically, they arise from an unrolled exact couple in which E^{s} = 0 for all s < 0, so that i: A^{s} ≅ A^{s−1} ≅ … ≅ A^{−∞} for all s ≤ 0, and this is the only case we discuss in detail. (Again, other unrolled exact couples can produce them. The results remain valid when appropriately modified, as all arguments can be carried out degreewise; the difficulty is to find notation that helps rather than hinders.)

Conditional convergence We state our main theorems.

Theorem 7.1 Suppose the unrolled exact couple (0.1) satisfies E^{s} = 0 for all s < 0 (equivalently, η^{s}: A^{s} ≅ A^{−∞} for all s ≤ 0), and that the resulting half-plane spectral sequence converges conditionally to the colimit A^{−∞} or the limit A^{∞} (see Definition 5.10). If RE_{∞} = 0, the spectral sequence converges strongly.

This theorem follows directly from the more detailed Theorems 7.3 and 7.4 for the two cases, which differ somewhat; both clearly demonstrate the role of the condition RE_{∞} = 0.

Remark It is often easy to verify that RE_{∞} = 0. This holds trivially if the spectral sequence collapses for any reason (d_{r} = 0 for all r ≥ 2, or merely for r ≥ r_{0}). More generally, it holds if, for each s and t, only finitely many of the differentials d_{r}: E_{r}^{s,t} → E_{r}^{s+r,t−r+1} are nonzero. This is certainly true if each E_{r}^{s,t} is finite for some r (which is allowed to depend on s and t) or satisfies a descending chain condition. These observations cover many of the standard applications.

In effect, this result divides the question of strong convergence into two parts. The first, conditional convergence, is a structural condition that holds for large classes of spectral sequences, regardless of the size or nature of the objects that appear. The second, RE_{∞} = 0, depends by definition only on data internal to the spectral sequence and cannot be expected to hold in general, unless we impose finiteness conditions so severe as to render all convergence questions moot.

If we have a morphism of such spectral sequences, we can hope for strong convergence so that Theorem 5.3 applies. We can do better, and obtain as good a result without it.

Theorem 7.2 Suppose given a morphism f: A^{s} → \overline{A}^{s} etc. of unrolled exact couples (0.1) in which E^{s} = \overline{E}^{s} = 0 for all s < 0, where the resulting spectral sequences converge conditionally to either (i) the colimits G = A^{−∞} and \overline{G} = \overline{A}^{−∞}, or (ii) the limits G = A^{∞} and \overline{G} = \overline{A}^{∞}. If f induces isomorphisms f_{∞}: E_{∞} ≅ \overline{E}_{∞} and RE_{∞} ≅ R\overline{E}_{∞}, then it induces an isomorphism f: G ≅ \overline{G} of filtered target groups (in the strict sense that f: F^{s}G ≅ F^{s}\overline{G} for all s). Further, f induces an isomorphism RA^{∞} ≅ R\overline{A}^{∞} in Case (ii) (as well as trivially in Case (i)).

Remark As in Theorem 2.6, one can show (with extra work) that the hypothesis R\overline{A}^{∞} = 0 in Case (i) is not needed; it is in fact a consequence.

Note that we do not assume that RE_{∞} = 0, so that we obtain the desired isomorphism even for spectral sequences that fail to be weakly convergent! This makes conditional convergence a tool for theoretical work as well as practical computations. Just as it is useful in analysis to consider divergent series that are nevertheless summable in some sense, this theorem makes it reasonable to consider certain spectral sequences that fail all forms of convergence listed in Definition 5.2.

Proof In Case (i), naturality of (5.12) yields the commutative diagram

[figure: 0; \frac{F^{s}G}{F^{s+1}G}; E_{∞}^{s}; Q^{s+1}; Q^{s}; RE_{∞}^{s}; 0; 0; \frac{F^{s}\overline{G}}{F^{s+1}\overline{G}}; \overline{E}_{∞}^{s}; \overline{Q}^{s+1}; \overline{Q}^{s}; R\overline{E}_{∞}^{s}; 0; ≅; i; f; f; ≅; i]

with exact rows. Put K^{s} = Ker[f: Q^{s} → \overline{Q}^{s}] and C^{s} = Coker[f: Q^{s} → \overline{Q}^{s}]; by Lemma 1.7, K^{s+1} → K^{s} is epic and C^{s+1} ≅ C^{s}. The left exactness of lim shows that lim_{s} K^{s} = 0, since lim_{s} Q^{s} = A^{∞} = 0; hence K^{s} = 0 for all s by Corollary 1.9.

We thus have the short exact sequence

0 ⟶ Q^{s} ⟶^{f} \overline{Q}^{s} ⟶ C^{s} ⟶ 0,

to which we apply lim_{s} by Theorem 1.4. Since RA^{∞} = 0 implies Rlim_{s} Q^{s} = 0 by Theorem 3.4 and lim_{s} \overline{Q}^{s} = \overline{A}^{∞} = 0, we deduce that lim_{s} C^{s} = 0. Then C^{s} = 0 for all s, and we have f: Q^{s} ≅ \overline{Q}^{s} for all s.

The above diagram now provides isomorphisms F^{s}G/F^{s+1}G ≅ F^{s}\overline{G}/F^{s+1}\overline{G} for all s. We apply Theorem 2.6 to the filtered homomorphism f: G → \overline{G}. Both filtrations are complete, by Lemma 5.11(b). The isomorphism η^{0}: A^{0} ≅ A^{−∞} carries Q^{0} to F^{∞}, and similarly for \overline{A}^{0}, so that f: F^{∞} ≅ \overline{F}^{∞}.

In Case (ii), we obtain from (5.7) the similar commutative diagram

[figure: 0; E_{∞}^{s}; Q^{s+1}; Q^{s}; RE_{∞}^{s}; RQ^{s+1}; RQ^{s}; 0; 0; \overline{E}_{∞}^{s}; \overline{Q}^{s+1}; \overline{Q}^{s}; R\overline{E}_{∞}^{s}; R\overline{Q}^{s+1}; R\overline{Q}^{s}; 0; ≅; i; f; f; ≅; i]

This time, we find an isomorphism K^{s+1} ≅ K^{s} and a monomorphism C^{s+1} → C^{s}. Since A^{0} = \overline{A}^{0} = 0, we must have K^{s} = C^{s} = 0 for all s, and we again deduce isomorphisms f: Q^{s} ≅ \overline{Q}^{s} for all s.

Hence f: Im^{α} A^{s} ≅ Im^{α} \overline{A}^{s} for all α ≥ ω and all s. Then diagram (5.8) yields the isomorphism F^{s}G/F^{s+1}G ≅ F^{s}\overline{G}/F^{s+1}\overline{G}, and Lemma 5.4(b) allows us to apply Theorem 2.6 to f: G → \overline{G}.

There is more. Put L^{s} = Ker[RQ^{s} → R\overline{Q}^{s}] and D^{s} = Coker[RQ^{s} → R\overline{Q}^{s}]; Lemma 1.7 gives L^{s+1} ≅ L^{s} and D^{s+1} ≅ D^{s} for all s. We again have L^{0} = D^{0} = 0 for trivial reasons, and so L^{s} = D^{s} = 0 and RQ^{s} ≅ R\overline{Q}^{s} for all s. Then naturality of the Mittag–Leffler short exact sequence (3.5) yields RA^{∞} ≅ R\overline{A}^{∞}.

The colimit as target We can be more specific when the target G is the colimit A^{−∞}. The conditions in Theorem 7.1 are exactly what we need.

Theorem 7.3 Suppose that the unrolled exact couple (0.1) satisfies E^{s} = 0 for all s < 0, so that we have a half-plane spectral sequence with entering differentials. If the spectral sequence satisfies any two of the following conditions, it satisfies the third:

(i) The spectral sequence converges conditionally to the colimit A^{−∞}, (i. e. A^{∞} = 0 and RA^{∞} = 0);

(ii) RE_{∞} = 0; (iii) The spectral sequence converges strongly to A^{−∞}.

Proof The filtration of G = A^{−∞} is trivially exhaustive, since F^{0}G = G. Further, the filtered isomorphism η^{0}: A^{0} ≅ G induces Q^{0} ≅ F^{∞} and RQ^{0} ≅ RF^{∞}.

If we assume (i) and (ii), Lemma 5.9 yields isomorphisms RF^{∞} ≅ RQ^{0} ≅ RA^{∞} = 0 and weak convergence, and hence F^{∞} ≅ Q^{0} ≅ A^{∞} = 0. We have (iii).

Conversely, if we assume (ii) and (iii), the same isomorphisms yield RA^{∞} ≅ RF^{∞} = 0 and A^{∞} ≅ F^{∞} = 0, and we have (i).

If we assume (i) and weak convergence, Lemma 5.11(c) shows that i: Q^{s+1} → Q^{s} is monic for all s. But (iii) gives Q^{0} ≅ F^{∞} = 0, so that we must have Q^{s} = 0 for all s. Then diagram (5.12) reduces to (ii).

Example We exhibit a conditionally convergent half-plane spectral sequence for which RE_{∞} ≠ 0. We take A^{s} = ℤ̂_{(2)} for s > 0 and A^{0} = A^{−∞} = G = ℤ̂_{(2)}/ℤ, all concentrated in degree 0. For s > 0, i: A^{s+1} → A^{s} is multiplication by 2, and i: A^{1} → A^{0} is the natural projection. Then F^{s}G = G for all s. Hence Q^{0} = F^{∞} = G, but Q^{s} = 0 for s > 0, so that RE_{∞}^{0} ≅ Q^{0} = G from (5.12).

§8 Whole-plane spectral sequences 23

We take deg(j) = 0 and deg(k) = 1, as usual. From (0.1), the only nonzero groups in the E_{1}-term are E_{1}^{0,−1} = ℤ, generated by x, say, and E_{1}^{s,−s} = ℤ/2, for s > 0, generated by y_{s}. Then E_{r}^{0,−1} = ℤ, generated by 2^{r−1}x, with differential d_{r}(2^{r−1}x) = y_{r}. Thus E_{∞} = 0 and we trivially have weak convergence.

The limit as target The other case is rather different.

Theorem 7.4 Suppose A^{s} = 0 for all s ≤ 0 in the unrolled exact couple (0.1), so that E^{s} = 0 for all s < 0 and A^{−∞} = 0. Then the following conditions on the resulting half-plane spectral sequence are equivalent:

(i) RE_{∞} = 0; (ii) RA^{∞} = 0 and the spectral sequence converges strongly to A^{∞}.

Proof We trivially have Q^{0} = RQ^{0} = 0. We combine diagrams (5.7) and (5.8) into the commutative diagram with exact rows (omitting 0 from each end)

[figure: \frac{F^{s}G}{F^{s+1}G}; Im^{σ} A^{s+1}; Im^{σ} A^{s}; 0; E_{∞}^{s}; Q^{s+1}; Q^{s}; RE_{∞}^{s}; RQ^{s+1}; RQ^{s}; ⊂; ⊂; i]  (7.5)

If RE_{∞} = 0, Lemma 5.9 gives RA^{∞} ≅ RQ^{s} ≅ RQ^{0} = 0 and Im^{σ} A^{s} = Q^{s}. The diagram yields weak (therefore, by Lemma 5.4(b), strong) convergence.

Conversely, suppose we have convergence. The diagram shows, by Lemma 1.7, that Im^{σ} A^{s} = Q^{s} implies Im^{σ} A^{s+1} = Q^{s+1}; by induction, starting with Q^{0} = 0, we deduce that Im^{σ} A^{s} = Q^{s} for all s. By Corollary 3.6, RA^{∞} = 0 implies that RQ^{s} = 0 for all s. The diagram now shows that RE_{∞}^{s} = 0.

8 Whole-plane spectral sequences

In this section, we extend our previous results to spectral sequences arising from a general unrolled exact couple with no dimensional restrictions.

Another obstruction The condition RE_{∞} = 0 is no longer enough to guarantee strong convergence; an additional obstruction group W arises from the interaction between limits and colimits. We introduce W formally later, in equation (8.7); meanwhile, we give a useful criterion for it to vanish, that depends only on data internal to the spectral sequence. In stating the criterion, we assume that deg(i) = deg(j) = 0 and deg(k) = 1, and give the differentials the usual (cohomological) indexing, d_{r}: E_{r}^{s,t} → E_{r}^{s+r,t−r+1}. (It can of course be adapted to other conventions.)

Lemma 8.1 Suppose that for each m, there exist numbers u(m) and v(m) such that for all u ≥ u(m) and v ≥ v(m), the differential

d_{u+v}: E_{u+v}^{−u,m+u} ⟶ E_{u+v}^{v,m−v+1}

vanishes. Then W = 0.

Remark This criterion has a convenient graphical interpretation. As usual, we place the group E_{r}^{s,t} at the point (s, t) of the plane. We represent each differential d_{r}: E_{r}^{s,t} → E_{r}^{s+r,t−r+1} by a line segment joining the source (s, t) and the target (s+r, t−r+1). Then the hypothesis states that there does not exist an infinite family of nonzero differentials that all cross each other (in their interiors).

Conditional convergence We generalize Theorems 7.1 and 7.2.

Theorem 8.2 Suppose the spectral sequence arising from the unrolled exact couple (0.1) converges conditionally to the colimit A^{−∞} or the limit A^{∞}. If RE_{∞} = 0 and W = 0, the spectral sequence converges strongly.

This will follow immediately from the more detailed Theorems 8.10 and 8.13.

Theorem 8.3 Assume given unrolled exact couples A^{s} etc. and \overline{A}^{s} etc. as in (0.1) that converge conditionally to either (i) the colimits G = A^{−∞} and \overline{G} = \overline{A}^{−∞} or (ii) the limits G = A^{∞} and \overline{G} = \overline{A}^{∞}. Suppose f: A^{s} → \overline{A}^{s} etc. is a morphism of unrolled exact couples that induces isomorphisms:

(i) f_{∞}: E_{∞} ≅ \overline{E}_{∞};
(ii) RE_{∞} ≅ R\overline{E}_{∞} (e. g. if RE_{∞} = R\overline{E}_{∞} = 0);
(iii) W ≅ \overline{W} (e. g. if W = \overline{W} = 0).

Then f induces an isomorphism of filtered groups f: G ≅ \overline{G} (in the strict sense that f: F^{s}G ≅ F^{s}\overline{G} for all s).

Again, we defer the proof. In fact, we have to treat the two cases separately.

Limits and colimits in a sequence The new feature of whole-plane spectral sequences is that limits and colimits can interact in complicated and mysterious ways. In addition to the decreasing filtration of each A^{s} by the images Im^{r} A^{s} = Im[i^{(r)}: A^{s+r} → A^{s}], we have the increasing filtration by the kernels K_{n}A^{s} = Ker[i^{(n)}: A^{s} → A^{s−n}]. We define K_{n} Im^{r} A^{s} = K_{n}A^{s} ∩ Im^{r} A^{s}. The minimal subquotient groups of this double filtration are directly expressible in terms of the spectral sequence.

Lemma 8.4 For any n ≥ 1, r ≥ 0, and s, we have the natural isomorphism

\frac{K_{n} Im^{r} A^{s}}{K_{n} Im^{r+1} A^{s} + K_{n−1} Im^{r} A^{s}} ≅ Im[d_{n+r}: E_{n+r}^{s−n} → E_{n+r}^{s+r}].

(If deg(i) = deg(j) = 0, this isomorphism has degree 0.)

Proof The homomorphism i^{(n−1)}: A^{s} → A^{s−n+1} carries the stated subquotient isomorphically to K_{1} Im^{r+n−1} A^{s−n+1}/K_{1} Im^{r+n} A^{s−n+1}, which equation (0.7) identifies with Im d_{n+r}.

In addition to Q^{s} = Im^{ω} A^{s} = lim_{r} Im^{r} A^{s} and RQ^{s} = Rlim_{r} Im^{r} A^{s}, we introduce K_{∞}A^{s} = Ker[η^{s}: A^{s} → G] = ⋃_{n} K_{n}A^{s}, and extend our previous notation K_{n} Im^{r} A^{s} to allow r = ω or n = ∞ or both. These behave exactly as expected under limits (intersections) and colimits (unions), namely

⋂_{r} K_{n} Im^{r} A^{s} = ⋂_{r}(K_{n}A^{s} ∩ Im^{r} A^{s}) = K_{n}A^{s} ∩ Q^{s} = K_{n}Q^{s},

even for n = ∞, and

⋃_{n} K_{n} Im^{r} A^{s} = ⋃_{n}(K_{n}A^{s} ∩ Im^{r} A^{s}) = K_{∞}A^{s} ∩ Im^{r} A^{s} = K_{∞} Im^{r} A^{s},

even for r = ω. Thus these limits and colimits commute,

⋂_{r} ⋃_{n} K_{n} Im^{r} A^{s} = K_{∞}Q^{s} = ⋃_{n} ⋂_{r} K_{n} Im^{r} A^{s}.

However, the derived limits do not commute with colimits in general.

Lemma 8.5 There is a short exact sequence

0 ⟶ colim_{n} Rlim_{r} K_{n} Im^{r} A^{s} ⟶ Rlim_{r} colim_{n} K_{n} Im^{r} A^{s} ⟶ W ⟶ 0, (8.6)

where

W = colim_{s} Rlim_{r} K_{∞} Im^{r} A^{s}.  (8.7)

In terms of the double filtration on A^{s}, we can write

W = colim_{n} Rlim_{r} \frac{K_{∞} Im^{r} A^{s}}{K_{n} Im^{r} A^{s}}  (8.8)

Proof We apply lim_{r} by Theorem 1.4 to the short exact sequence

0 ⟶ K_{n} Im^{r} A^{s} ⟶^{⊂} K_{∞} Im^{r} A^{s} ⟶^{i^{(n)}} K_{∞} Im^{r+n} A^{s−n} ⟶ 0  (8.9)

to obtain (in part) the exact sequence

K_{∞}Q^{s−n} ⟶ Rlim_{r} K_{n} Im^{r} A^{s} ⟶ Rlim_{r} K_{∞} Im^{r} A^{s} ⟶ Rlim_{r} K_{∞} Im^{r+n} A^{s−n} ⟶ 0.

To this we apply the exact functor colim_{n}. Since

colim_{n} K_{∞}Q^{s−n} = Ker[colim_{n} Q^{s−n} → G] = colim_{n} Ker[Q^{s−n} → A^{s−n}] = 0,

we obtain (8.6), with W = colim_{n} Rlim_{r} K_{∞} Im^{r+n} A^{s−n}. By shifting the indexing (which does not affect the colimit or derived limit) and relabeling, we find the form (8.7) of W.

For (8.8), we use diagram (8.9) to rewrite K_{∞} Im^{r+n} A^{s−n} as a quotient.

We deduce Lemma 8.1 by combining Lemmas 8.4 and 8.5.

Proof of Lemma 8.1 We work throughout in degree m+1. In Lemma 8.4 we set s = −u(m)+1, n = u−u(m)+1, and r = v+u(m)−1, and use it to rewrite the hypothesis of Lemma 8.1 as

K_{n} Im^{r} A^{s} = K_{n} Im^{r+1} A^{s} + K_{n−1} Im^{r} A^{s}

for all n ≥ 1 and all r ≥ r_{0} = u(m)+v(m)−1. Induction on n, starting from the triviality K_{0}A^{s} = 0, shows that K_{n} Im^{r} A^{s} = K_{n} Im^{r+1} A^{s} for all r ≥ r_{0} and all finite n, hence also for n = ∞. Thus K_{∞} Im^{r} A^{s}/K_{n} Im^{r} A^{s} is independent of r for r ≥ r_{0}, and equation (8.8) shows that W = 0.

The colimit as target Here, we take the colimit A^{−∞} as the target G of our spectral sequence, filtered by the subgroups F^{s}G = F^{s}A^{−∞} = Im[η^{s}: A^{s} → A^{−∞}].

Theorem 8.10 If the spectral sequence resulting from the unrolled exact couple (0.1) satisfies any two of the following conditions, it satisfies the third:

(i) The spectral sequence converges conditionally to the colimit A^{−∞} (i. e. A^{∞} = 0 and RA^{∞} = 0); (ii) RE_{∞} = 0 and W = 0; (iii) The spectral sequence converges strongly to A^{−∞}.

By Lemma 5.4(a), the filtration of A^{−∞} is always exhaustive. We need to study the groups F^{∞} and RF^{∞}. The isomorphisms Q^{0} ≅ F^{∞} and RQ^{0} ≅ RF^{∞} that were so useful in §7 do not extend in the obvious way; the natural homomorphisms colim_{s} Q^{s} → F^{∞} and colim_{s} RQ^{s} → RF^{∞} are in general not isomorphisms.

Lemma 8.11 There is an exact sequence

0 ⟶ colim_{s} Q^{s} ⟶ F^{∞} ⟶ W ⟶ colim_{s} RQ^{s} ⟶ RF^{∞} ⟶ 0.  (8.12)

Proof We apply lim_{r} by Theorem 1.4 to the short exact sequence

0 ⟶ K_{∞} Im^{r} A^{s} ⟶^{⊂} Im^{r} A^{s} ⟶^{η^{s}} F^{r+s} ⟶ 0

to obtain the exact sequence

K_{∞}Q^{s} ⟶ Q^{s} ⟶ F^{∞} ⟶ Rlim_{r} K_{∞} Im^{r} A^{s} ⟶ RQ^{s} ⟶ RF^{∞} ⟶ 0.

To this we apply the exact functor colim_{s}; as in the proof of Lemma 8.5, we see that colim_{s} K_{∞}Q^{s} = 0. We deduce (8.12), with the help of equation (8.7).

Proof of Theorem 8.10 We modify the proof of Theorem 7.3.

Assume (i) and (ii). Lemma 5.9 yields weak convergence, also Q^{s} = RQ^{s} = 0 for all s. Since W = 0, Lemma 8.11 gives F^{∞} = RF^{∞} = 0, and we have (iii).

Conversely, (ii) and (iii) imply (i).

As before, (i) and (iii) imply monomorphisms i: Q^{s+1} → Q^{s} and RQ^{s} = 0 for all s. Since F^{∞} = RF^{∞} = 0, Lemma 8.11 shows that W = 0 and colim_{s} Q^{s} = 0. It follows that Q^{s} = 0 for all s. Then RE_{∞} = 0 by diagram (5.12).

Proof of Theorem 8.3, Case (i) The proof of Theorem 7.2 still provides isomorphisms f: Q^{s} ≅ \overline{Q}^{s} and F^{s}/F^{s+1} ≅ \overline{F}^{s}/\overline{F}^{s+1}. By Lemma 5.11, RQ^{s} = R\overline{Q}^{s} = RF^{∞} = R\overline{F}^{∞} = 0. Then naturality of diagram (8.12) shows that f: F^{∞} ≅ \overline{F}^{∞}, and Theorem 2.6 applies to f: G → \overline{G}.

Example We exhibit an unrolled exact couple for which W ≠ 0. For integers n ≥ 0, we take A^{n} and A^{−n} to be free abelian on generators x_{t} in degree zero for all t ≥ n and i: A^{s+1} → A^{s} as the obvious homomorphism given by ix_{t} = x_{t} for all t, except that i: A^{−n} → A^{−n−1} takes x_{n} to x_{n+1}. Then G = A^{−∞} is free on one generator x, with η^{s}x_{t} = x for all s and t. Equivalently, G = A^{0}/K, where K denotes the subgroup consisting of all finite sums ∑_{t} λ_{t}x_{t} such that ∑_{t} λ_{t} = 0. It is filtered trivially, with F^{s} = G for all s, so that F^{∞} = G ≠ 0 and RF^{∞} = 0. Clearly, Q^{s} = 0 for all s, and Lemma 8.11 shows that W ≠ 0. Also, RQ^{s} ≠ 0.

As usual, we take deg(i) = deg(j) = 0 and deg(k) = 1. Then E^{n} is free on one element z_{n} = jx_{n} in degree 0, and E^{−n−1} is free on one element y_{n} in degree −1, where ky_{n} = x_{n} − x_{n+1}. The differentials d_{2n+1}: E_{2n+1}^{−n−1,n} → E_{2n+1}^{n,−n} in the spectral sequence are given by d_{2n+1}y_{n} = z_{n} and kill everything, so that E_{∞} = RE_{∞} = 0. It is clear how the hypothesis of Lemma 8.1 fails.

To obtain a conditionally convergent example for Theorem 8.10, we complete the sequence A as in the proof of Theorem 3.4, so that Â^{n} = Â^{−n} consists of formal infinite sums ∑_{t} λ_{t}x_{t}. This leaves E^{s} and the spectral sequence undisturbed, but we now have conditional convergence to Â^{0}/K. (This is not a general phenomenon; in view of diagram (3.8) and Lemma 1.7, the procedure works precisely when Q^{s+1} ≅ Q^{s} and RQ^{s+1} ≅ RQ^{s} for all s.) The filtration of the target group Â^{−∞} is trivial as before, and not Hausdorff. Thus Theorem 8.10 fails without the hypothesis W = 0.

The limit as target Here, we assume A^{−∞} = 0 and take the limit A^{∞} as the target G, filtered by the subgroups F^{s} = F^{s}G = Ker[ϵ^{s}: A^{∞} → A^{s}]. By Lemma 5.4(b), this filtration is always complete Hausdorff and exhaustive.

Theorem 8.13 Assume only that A^{−∞} = 0 in the unrolled exact couple (0.1). Then the following conditions on the resulting spectral sequence are equivalent:

(i) RE_{∞} = 0 and W = 0; (ii) RA^{∞} = 0 and the spectral sequence converges strongly to A^{∞}.

In this context, the mystery group W has a simpler interpretation.

Lemma 8.14 If A^{−∞} = 0, then W = colim_{s} RQ^{s}.

Proof In this case, K_{∞} Im^{r} A^{s} = Im^{r} A^{s}, and equation (8.7) simplifies.

Proof of Theorem We modify the proof of Theorem 7.4.

As before, (i) implies strong convergence, also that RA^{∞} ≅ RQ^{s} for all s. Then Lemma 8.14 shows that RA^{∞} ≅ W = 0.

If we assume (ii), we still have diagram (7.5), but the inductive proof we used before is no longer available. Instead, put C^{s} = Q^{s}/ Im^{σ} A^{s}; then by Lemma 1.7, C^{s+1} → C^{s} is monic. But colim_{s} C^{s} = 0, being a quotient of colim_{s} Q^{s} = 0; we deduce that C^{s} = 0 and Im^{σ} A^{s} = Q^{s} for all s. By Corollary 3.6, RQ^{s} = 0 for all s. Then W = 0 by Lemma 8.14 and RE_{∞}^{s} = 0 by diagram (7.5).

Proof of Theorem 8.3, Case (ii) We modify the proof of Theorem 7.2, by making much use of colimits and their exactness.

We still have K^{s+1} ≅ K^{s} and monomorphisms C^{s+1} → C^{s}, where K^{s} = Ker[f: Q^{s} → \overline{Q}^{s}] and C^{s} = Coker[f: Q^{s} → \overline{Q}^{s}]. We no longer have Q^{0} = \overline{Q}^{0} = 0. Instead, we use colim_{s} Q^{s} = 0 and colim_{s} \overline{Q}^{s} = 0; these are subgroups of A^{−∞} = 0 and \overline{A}^{−∞} = 0. Then colim_{s} K^{s} = colim_{s} C^{s} = 0, and we conclude that K^{s} = C^{s} = 0 for all s. As before, we deduce the filtered isomorphism f: G ≅ \overline{G}.

§9 Filtered complexes 28

We still find L^{s+1} ≅ L^{s} and D^{s+1} ≅ D^{s}, where L^{s} = Ker[RQ^{s} → R\overline{Q}^{s}] and D^{s} = Coker[RQ^{s} → R\overline{Q}^{s}]. This time, we use Lemma 8.14 to write colim_{s} L^{s} = Ker[W → \overline{W}] = 0 and colim_{s} D^{s} = Coker[W → \overline{W}] = 0. It follows that L^{s} = D^{s} = 0 for all s, and we finish the proof as before.

Part III — EXAMPLES

We give examples of unrolled exact couples and the resulting spectral sequences in which the concept of conditional convergence is relevant. A common theme will be to map a given unrolled exact couple to or from another one having better properties, by a morphism that preserves the groups E^{s} and therefore the whole spectral sequence, including the infinite terms (5.1).

9 Filtered complexes

Given a (chain) complex C equipped with a decreasing filtration by subcomplexes F^{s}C, we construct an unrolled exact couple and hence a spectral sequence. The homology exact sequence of the short exact sequence of complexes

0 ⟶ F^{s+1}C ⟶^{i} F^{s}C ⟶^{p} F^{s}C/F^{s+1}C ⟶ 0

is the required long exact sequence. We define the unrolled exact couple by

A^{s} = H(F^{s}C),  E^{s} = H(F^{s}C/F^{s+1}C),  (9.1)

together with the obvious structure maps H(i): A^{s+1} → A^{s}, j = H(p): A^{s} → E^{s}, and the connecting homomorphism k = δ: E^{s} → A^{s+1}.

We impose no dimensional restrictions on C or the filtration. If F^{m}C = 0 for some finite m, we find a half-plane spectral sequence with exiting differentials. If F^{m}C = C for some finite m, we find a half-plane spectral sequence with entering differentials. Typically, the differential ∂ of C has degree +1, but it is often convenient to reindex C or the filtration.

Theorem 9.2 If the filtration of C by the subcomplexes F^{s}C exhausts C and is complete Hausdorff, then the spectral sequence resulting from (9.1) is

E_{1}^{s} = H(F^{s}C/F^{s+1}C) ⟹ H(C),

and it converges conditionally to the colimit H(C).

Proof By exactness of the colimit functor, the target group is

A^{−∞} = colim_{s} H(F^{s}C) ≅ H(colim_{s} F^{s}C) = H(C).

We have to show that A^{∞} = RA^{∞} = 0, which is essentially Proposition 6.1 of Eilenberg–Moore [5]. By hypothesis, diagram (1.3) reduces to the isomorphism

1 − i: ∏_{s} F^{s}C ⟶^{≅} ∏_{s} F^{s}C.

§10 Double complexes 29

We apply homology, which preserves products, to obtain another isomorphism

1 − H(i): ∏_{s} H(F^{s}C) ⟶^{≅} ∏_{s} H(F^{s}C),

which, again by reference to diagram (1.3), gives the result.

One often needs to consider general filtered complexes. If the subcomplexes F^{s}C do not exhaust C it is obvious what to do: we simply replace C by F^{−∞}C = ⋃_{s} F^{s}C. If the filtration is not complete Hausdorff, Proposition 2.8 constructs the completion Ĉ, filtered by subgroups F^{s}Ĉ, with a differential provided by naturality.

Theorem 9.3 Suppose the complex C is filtered by subcomplexes F^{s}C that exhaust C. Then we have the spectral sequence

E_{1}^{s} = H(F^{s}C/F^{s+1}C) ⟹ H(Ĉ),

which converges conditionally to the colimit H(Ĉ), where Ĉ is the completion of C.

Proof By Proposition 2.8(b), the completion homomorphism C → Ĉ induces isomorphisms F^{s}C/F^{s+1}C ≅ F^{s}Ĉ/F^{s+1}Ĉ, and therefore an isomorphism of spectral sequences. We apply Theorem 9.2 to Ĉ.

10 Double complexes

A double complex D is a bigraded group with components D^{s,t} having degree s+t, equipped with two differentials ∂′: D^{s,t} → D^{s+1,t} and ∂′′: D^{s,t} → D^{s,t+1} of degree +1 that anticommute, ∂′′ ^{∘} ∂′ = −∂′ ^{∘} ∂′′. (Again, it is often convenient to change the indexing.) We construct its spectral sequence.

The total complex C of D has the component C^{n} = ⨁_{s+t=n} D^{s,t} in degree n and total differential ∂ = ∂′ + ∂′′: C^{n} → C^{n+1}. We filter C by the subcomplexes F^{s}C, where F^{s}C^{n} = ⨁_{i≥s} D^{i,n−i}. The spectral sequence of the double complex D is by definition the spectral sequence of this filtered complex C. (More precisely, this is one of them — there is a second spectral sequence with the roles of ∂′ and ∂′′ interchanged.) If D^{s,t} = 0 for all s > s_{0} (resp. s < s_{0}) and all t, it is a half-plane spectral sequence with exiting (resp. entering) differentials.

Then F^{s}C/F^{s+1}C is just the complex (D^{s,∗}, ∂′′), with D^{s,t} in degree s + t, so that E_{1}^{s} = H(D^{s,∗}, ∂′′). Next, one computes that d_{1}: E_{1}^{s} → E_{1}^{s+1}, a connecting homomorphism, is the homology homomorphism induced by the morphism ∂′: D^{s,∗} → D^{s+1,∗} of ∂′′-complexes of degree +1. This gives the E_{2}-term.

The filtration of C clearly exhausts C and is Hausdorff, but in general fails to be complete. We therefore apply Theorem 9.3 rather than Theorem 9.2.

Theorem 10.1 The spectral sequence of the double complex D,

E_{2} = H(H(D, ∂′′), H(∂′)) ⟹ H(Ĉ),

converges conditionally to the colimit H(Ĉ), where Ĉ denotes the completion of the total complex of D.

§11 Multicomplexes 30

The completion in this context is readily constructed. The group Ĉ^{n} consists of those elements x ∈ ∏_{s} D^{s,n−s} whose components x_{s} ∈ D^{s,n−s} vanish for all s < s_{0}, for some s_{0} that depends on x. (Note that in general, this is neither the sum ⨁_{s} D^{s,n−s} nor the product ∏_{s} D^{s,n−s}.) In practice, it often happens that for dimensional reasons, only finitely many of the x_{s} can be nonzero, in which case C was already complete.

11 Multicomplexes

Multicomplexes are a flexible generalization of double complexes that were introduced by Wall [15] and are particularly useful in homological algebra. A multicomplex M consists of a bigraded group with components M^{s,t} having degree s + t and homomorphisms d_{r}: M^{s,t} → M^{s+r,t−r+1} of degree +1 for all r ≥ 0 that satisfy the identities

∑_{i+j=n} d_{i} ^{∘} d_{j} = 0: M^{s,t} ⟶ M^{s+n,t−n+2}  (11.1)

for all n ≥ 0 and all s and t. In particular, d_{0} is a differential. In the special case when d_{r} = 0 for all r ≥ 2, M is precisely a double complex.

As in §10, the total complex C of M has C^{n} = ⨁_{s+t=n} M^{s,t} in degree n, with total differential ∂ = ∑_{r=0}^{∞} d_{r}: C^{n} → C^{n+1}; this works provided the following finiteness condition holds for all x ∈ M:

d_{r}x = 0 for all except finitely many r.  (11.2)

The relation (11.1) is exactly what is needed to make ∂ a differential. We filter C by F^{s}C^{n} = ⨁_{i≥s} M^{i,n−i}, and define the spectral sequence of the multicomplex M as the spectral sequence of this filtered complex. If M^{s,t} = 0 for all s > s_{0} (resp. s < s_{0}) and all t, it is a half-plane spectral sequence with exiting (resp. entering) differentials.

Again, F^{s}C/F^{s+1}C is the complex M^{s,∗} with differential d_{0}, so that E_{1}^{s} = H(M^{s,∗}, d_{0}). Relation (11.1) with n = 1 states that d_{1}: M^{s,∗} → M^{s+1,∗} is a morphism of d_{0}-complexes of degree +1, and H(d_{1}): H(M^{s,∗}, d_{0}) → H(M^{s+1,∗}, d_{0}) serves as d_{1}: E_{1}^{s} → E_{1}^{s+1}. Although d_{1} ^{∘} d_{1} ≠ 0 in M, relation (11.1) with n = 2 shows that d_{2} is a chain homotopy between d_{1} ^{∘} d_{1} and 0, so that H(d_{1})^{∘} H(d_{1}) = 0 and we can still describe E_{2} as H(H(M, d_{0}), H(d_{1})). In general, as the notation suggests, the differential d_{r}: E_{r}^{s} → E_{r}^{s+r} is induced by d_{r}: M^{s,∗} → M^{s+r,∗}, but only on those classes that contain elements x ∈ M^{s,∗} that satisfy d_{i}x = 0 for all i < r (which rarely happens). Once again, we apply Theorem 9.3; the filtration of C is clearly Hausdorff and exhaustive, but not complete in general.

Theorem 11.3 The spectral sequence of the multicomplex M,

E_{2} = H(H(M, d_{0}), H(d_{1})) ⟹ H(Ĉ),

is conditionally convergent to the colimit H(Ĉ), where Ĉ denotes the completion of the total complex of M.

§12 Atiyah–Hirzebruch spectral sequences 31

Remark This theorem makes no reference to any finiteness condition on M because none is needed. As in §10, Ĉ^{n} is the set of all elements x ∈ ∏_{s} M^{s,n−s} for which x_{s} = 0 for all s < s_{0}, for some s_{0}. The differential is given by

(∂x)_{s} = ∑_{r≥0} d_{r}x_{s−r},

where the sum is guaranteed to be finite, and the formula defines an element ∂x ∈ Ĉ^{n+1}. In contrast, the differential in C exists only in the presence of the finiteness condition (11.2).

12 Atiyah–Hirzebruch spectral sequences

For simplicity of exposition we assume X is a cw-complex or cw-spectrum. When X is a space, we may use either absolute or reduced (co)homology, provided we are consistent; when X is a spectrum, only the reduced kind is available.

Homology We filter X by its skeletons X_{s}. Given a spectrum M, the M-homology exact sequence of the pair (X_{s}, X_{s−1}),

⟶ M_{∗}(X_{s−1}) ⟶ M_{∗}(X_{s}) ⟶ M_{∗}(X_{s}, X_{s−1}) ⟶^{δ_{∗}} M_{∗}(X_{s−1}) ⟶ …,

gives rise (after some awkward reindexing) to the obvious unrolled exact couple

A^{s} = M_{∗}(X_{−s}),  E^{s} = M_{∗}(X_{−s}, X_{−s−1}),  (12.1)

using the above homomorphisms as i, j, and k. Thus deg(i) = deg(j) = 0, while k has degree +1 (or homology degree −1), as usual. The resulting spectral sequence is known as the Atiyah–Hirzebruch homology spectral sequence of X.

To make the indices more manageable, it is customary to rewrite E_{r}^{−s,−t} as E_{s,t}^{r}, which has homology degree s + t. Then E_{s}^{1} = M_{∗}(X_{s}, X_{s−1}) may be regarded as the group of s-chains on X with coefficients in the homotopy groups M_{t} of M (shifted), which leads to the description E_{s,t}^{2} = H_{s}(X; M_{t}).

To discuss convergence, we need to know A^{∞} and A^{−∞}. Theorem 4.2(a) gives the target A^{−∞} = M_{∗}(X). From now on, we assume that X is a space or connective spectrum, so that X_{−1} is empty or a point spectrum and A^{∞} = 0, and we have a half-plane spectral sequence with exiting differentials. Thus Theorem 6.1 applies. (Otherwise, if X is a nonconnective spectrum, we have a whole-plane spectral sequence with A^{∞} = lim_{s} M_{∗}(X_{−s}), which need not vanish.)

Theorem 12.2 Let X be a space or connective spectrum and M any spectrum. Then the Atiyah–Hirzebruch homology spectral sequence,

E_{s,t}^{2} = H_{s}(X; M_{t}) ⟹ M_{∗}(X),

is a strongly convergent half-plane spectral sequence with exiting differentials.

Cohomology The cohomology theory M^{∗}(−) has coefficient groups M^{t} ≅ M_{−t}. The cohomology exact sequences of the triples (X, X_{s}, X_{s−1}) furnish the triangles for the unrolled exact couple with

A^{s} = M^{∗}(X, X_{s−1}),  E^{s} = M^{∗}(X_{s}, X_{s−1}).  (12.3)

Thus deg(i) = deg(j) = 0 and deg(k) = 1, as usual. The resulting spectral sequence is known as the Atiyah–Hirzebruch cohomology spectral sequence for M^{∗}(X). As in homology, one interprets E_{1}^{s,t} = M^{s+t}(X_{s}, X_{s−1}) as the group of cellular s-cochains on X with coefficients in M^{t}, and deduces the E_{2}-term.

We continue to assume that X is a space or connective spectrum, so that the target group is G = M^{∗}(X), and we have a half-plane spectral sequence with entering differentials. (Otherwise, we have a whole-plane spectral sequence with target G = colim_{s} M^{∗}(X, X_{s}), which is quite mysterious.) In any case, this spectral sequence is conditionally convergent by Theorem 4.3(b).

Theorem 12.4 Let X be a space or connective spectrum and M any spectrum. Then the Atiyah–Hirzebruch cohomology spectral sequence,

E_{2}^{s,t} = H^{s}(X; M^{t}) ⟹ M^{∗}(X),

is a half-plane spectral sequence with entering differentials that converges conditionally to the colimit M^{∗}(X).

Another exact couple There is a more obvious unrolled exact couple that delivers the same spectral sequence. We take

\overline{A}^{s} = M^{∗}(X_{s−1}),  E^{s} = M^{∗}(X_{s}, X_{s−1}),  (12.5)

and use the cohomology exact sequences of the pairs (X_{s}, X_{s−1}) as the triangles. Its spectral sequence converges conditionally to the limit target group \overline{G} = \overline{A}^{∞} = lim_{s} M^{∗}(X_{s}), which may or may not be the same as M^{∗}(X). The homomorphisms δ^{∗}: \overline{A}^{s} → A^{s} form a morphism of unrolled exact couples that preserves each E^{s} and hence the whole spectral sequence, including the terms E_{∞} and RE_{∞}.

We compare these two rather different descriptions of the same spectral sequence. We prefer the first, because its target group G is the one we really want. As the filtration of G = M^{∗}(X) is automatically complete and exhaustive, strong convergence to G reduces to two conditions:

(i) Weak convergence, F^{s}/F^{s+1} ≅ E_{∞}^{s}; (ii) The filtration is Hausdorff, F^{∞} = 0.

By Theorem 7.3, RE_{∞} = 0 if and only if both hold.

The Milnor short exact sequence (4.4) may be written

0 ⟶ R\overline{A}^{∞} ⟶ M^{∗}(X) ⟶ \overline{A}^{∞} ⟶ 0.

It allows us to identify F^{∞} = R\overline{A}^{∞}, and to compare the filtrations of G and \overline{G} by means of the short exact sequences

0 ⟶ R\overline{A}^{∞} ⟶ F^{s} ⟶ \overline{F}^{s} ⟶ 0.

Thus F^{s}/F^{s+1} ≅ \overline{F}^{s}/\overline{F}^{s+1}, and weak convergence in either sense is the same, even when G ≠ \overline{G}. As the filtration of \overline{G} is automatically complete Hausdorff and exhaustive by Lemma 5.4, strong convergence to \overline{G} requires only weak convergence, i. e. condition (i) above. Then Theorem 7.4 also shows that RE_{∞} = 0 is equivalent to both conditions, since F^{∞} = R\overline{A}^{∞}.

Filtered spaces The only purpose of filtering X by skeletons was to recognize the E_{2}-term. For any filtration of X by subcomplexes or subspectra, we still have the unrolled exact couples (12.1) and (12.3) and the resulting spectral sequences.

§13 Serre spectral sequences of a fibration 33

Theorem 12.6 Suppose X has an increasing filtration by subspaces or subspectra X_{s} that exhaust X, with X_{−1} trivial. Then for any spectrum M:

(a) The homology spectral sequence resulting from (12.1),

E_{s,t}^{1} = M_{s+t}(X_{s}, X_{s−1}) ⟹ M_{∗}(X),

is a strongly convergent half-plane spectral sequence with exiting differentials; (b) The cohomology spectral sequence resulting from (12.3),

E_{1}^{s,t} = M^{s+t}(X_{s}, X_{s−1}) ⟹ M^{∗}(X),

is a half-plane spectral sequence with entering differentials that converges conditionally to the colimit M^{∗}(X).

13 Serre spectral sequences of a fibration

Let p: X → B be a fibration with fibre F. Again to simplify the exposition, we assume that B is a cw-complex. We sketch the spectral sequences following Spanier [13, Chap. 9], omitting most details. We filter X by the inverse images X_{s} = p^{−1}(B_{s}) of the skeletons B_{s} of B. We use the homology and cohomology theories defined by the spectrum M, which may well be ordinary (co)homology.

Homology The homology spectral sequence (see Theorem 12.6(a)) of the filtered space X is known as the Serre homology spectral sequence of the fibration p: X → B. One can identify E_{s,t}^{1} with the group of cellular s-chains on X with coefficients in M_{t}(F). However, to proceed to the E^{2}-term, one must take account of the action of the fundamental group π_{1}(B) on M_{t}(F).

Theorem 13.1 Given a fibration p: X → B as above and a spectrum M, the Serre homology spectral sequence,

E_{s,t}^{2} = H_{s}(X; M_{t}(F)) ⟹ M_{∗}(X),

(with coefficients twisted by the action of π_{1}(B) on M_{t}(F)) is a strongly convergent half-plane spectral sequence with exiting differentials.

Cohomology The cohomology spectral sequence (see Theorem 12.6(b)) of the filtered space X is known as the Serre cohomology spectral sequence of the fibration. We shall be even briefer.

Theorem 13.2 Given a fibration p: X → B as above and a spectrum M, the Serre cohomology spectral sequence,

E_{2}^{s,t} = H^{s}(X; M^{t}(F)) ⟹ M^{∗}(X)

(with twisted coefficients), is a half-plane spectral sequence with entering differentials that converges conditionally to the colimit M^{∗}(X).

§14 Bockstein spectral sequences 34

14 Bockstein spectral sequences

Let p be a fixed prime. We start from an exact couple of graded groups

[figure: A; A; E; p; j; k]  (14.1)

where p: A → A denotes multiplication by p. By exactness, E consists of torsion of orders p and p^{2} only. We form an unrolled exact couple (0.1) by setting A^{s} = A and E^{s} = E for all s. The resulting spectral sequence is our algebraic Bockstein spectral sequence; it will include the topological spectral sequences as special cases. It is a whole-plane spectral sequence, but because all columns are identical, only one column needs to be considered.

As usual, we take the target group G to be the colimit A^{−∞}, filtered by F^{s}G = Im[η^{s}: A^{s} → A^{−∞}]; as we shall see, it is often obvious or of little interest, and convergence is the real issue. Because E is already p-local, we may localize the exact couple at p by tensoring with ℤ_{(p)}, without affecting the spectral sequence. So without loss of generality, we may assume A is p-local if we wish.

Recall that a group is reduced if it contains no nontrivial divisible subgroup. Denote by 𝔽_{p} the field with p elements.

Lemma 14.2 Let P be the p-primary torsion subgroup of A. (If A is p-local, this is the whole torsion subgroup Tors(A) of A.) Then in the algebraic Bockstein spectral sequence resulting from (14.1):

(a) A^{−∞} = A ⊗ ℤ[p^{−1}]. (If A is p-local, this is the same as A ⊗ ℚ.)

(b) F^{s} ≅ A/P.

(c) F^{s}/F^{s+1} ≅ (A/P) ⊗ 𝔽_{p} ≅ (A/ Tors(A)) ⊗ 𝔽_{p}.

(d) A^{∞} ≅ lim_{s} p^{σ}A, where p^{σ}A is the largest subgroup of A that is divisible by p and we use the homomorphisms p: p^{σ}A → p^{σ}A. If P is reduced, A^{∞} ≅ p^{σ}A.

(e) ϵ^{s}: A^{∞} → A^{s} is monic if and only if P is reduced.

(f) If the p-adic filtration of A is complete Hausdorff, then the spectral sequence converges conditionally to the colimit A^{−∞}.

Proof For (a), we rewrite p: A → A as 1 ⊗ p: A ⊗ ℤ → A ⊗ ℤ. Because A ⊗ − preserves colimits, we have A^{−∞} = A ⊗ colim_{s} ℤ = A ⊗ ℤ[p^{−1}]. For (b), we note that Ker[η^{s}: A^{s} → A^{−∞}] = P; then (c) follows.

Clearly, Im^{r} A^{s} = p^{r}A^{s}, and the image filtration of A coincides with the p-adic filtration. Further, Im^{α} A^{s} = p^{α}A^{s} for all ordinals α, and p^{σ}A^{s} is as asserted in (d). Now ϵ^{s}: A^{∞} → p^{σ}A is always epic by (3.3), and is monic precisely when p^{σ}A contains no elements of order p. Since P is p-local, the largest divisible subgroup of P is P ∩ p^{σ}A. This gives the rest of (d) and also (e).

In (f), we have Q^{s} = RQ^{s} = 0. Then A^{∞} = RA^{∞} = 0 by Theorem 3.4.

Now we can apply Lemma 5.9.

Theorem 14.3 Assume that RE_{∞} = 0 in the algebraic Bockstein spectral sequence resulting from (14.1). Then we have weak convergence, E_{∞}^{s} ≅ (A/ Tors(A)) ⊗ 𝔽_{p}, if and only if the p-primary torsion subgroup of A is reduced.

One variant of the Bockstein spectral sequence truncates the unrolled exact couple by setting E^{s} = 0 for all s < 0 (and i = 1: A^{s} → A^{s−1} for all s ≤ 0); the resulting spectral sequence is now a half-plane spectral sequence with entering differentials, and its columns are not all identical. The first three parts of Lemma 14.2 are modified in the obvious way to A^{−∞} = A, F^{s} = p^{s}A, and F^{s}/F^{s+1} = p^{s}A/p^{s+1}A (for s ≥ 0); the other three parts are unaffected.

Homology All the topological versions depend ultimately on the exact triangle

S ⟶^{p} S ⟶^{j} L(𝔽_{p}) ⟶^{k} S

of spectra. The composite β = j^{∘} k: L(𝔽_{p}) → L(𝔽_{p}) is the Bockstein map. If we smash with a general spectrum M, we obtain another exact triangle

M ⟶^{p} M ⟶^{j} M ∧ L(𝔽_{p}) ⟶^{k} M,

where M ∧L(𝔽_{p}) defines the homology theory M_{∗}(−; 𝔽_{p}), known as M-homology with coefficients 𝔽_{p}. Given any space or spectrum X, we deduce the exact couple (14.1) with A = M_{∗}(X) and E = M_{∗}(X; 𝔽_{p}); the resulting spectral sequence is the homology Bockstein spectral sequence of X. Clearly, the differential d_{1}: E_{1}^{s} → E_{1}^{s+1} is induced by the Bockstein β. We may apply Theorem 14.3.

Theorem 14.4 Assume that RE_{∞} = 0 in the homology Bockstein spectral sequence for X. Then we have weak convergence, E_{∞}^{s} ≅ (M_{∗}(X)/ Tors(M_{∗}(X))) ⊗ 𝔽_{p}, if and only if the p-primary torsion subgroup of M_{n}(X) is reduced for all n.

The hypothesis on M_{n}(X) obviously holds if the group M_{n}(X) is finitely generated, which is the case if M and X are connective and of finite type. (We say that a spectrum X has finite type if it is connective and π_{n}(X) (or equivalently H_{n}(X)) is finitely generated for all n.)

Cohomology For the cohomology Bockstein spectral sequence of X, we take A = M^{∗}(X) and E = M^{∗}(X; 𝔽_{p}) in (14.1). Again, we apply Theorem 14.3.

Theorem 14.5 Assume that RE_{∞} = 0 in the cohomology Bockstein spectral sequence of X. Then we have weak convergence, E_{∞}^{s} ≅ (M^{∗}(X)/ Tors(M^{∗}(X))) ⊗ 𝔽_{p}, if and only if the p-primary torsion subgroup of M^{n}(X) is reduced for all n.

In the case of ordinary cohomology, where M is the Eilenberg–MacLane spectrum H(ℤ), we often know in advance that M^{n}(X) is finitely generated and hence reduced. For general M, this condition is far less accessible; we therefore offer an alternative criterion that does not depend on detailed knowledge of M^{n}(X).

Theorem 14.6 Suppose the p-adic filtration of the coefficient group M^{n} is complete Hausdorff for all n. Then the cohomology Bockstein spectral sequence is strongly convergent if and only if RE_{∞} = 0 and W = 0.

§15 Adams spectral sequences 36

Proof As in §4, we construct the homotopy limit N of the sequence of spectra

… ⟶^{p} M ⟶^{p} M ⟶^{p} M.

For it, we have the two Milnor short exact sequences (4.10),

0 ⟶ Rlim_{s} M^{n−1} ⟶ N^{n} ⟶ lim_{s} M^{n} ⟶ 0

and

0 ⟶ Rlim_{s} M^{n−1}(X) ⟶ N^{n}(X) ⟶ lim_{s} M^{n}(X) ⟶ 0.

By Lemma 14.2(f), with A = M^{∗} and E = M^{∗}(L(𝔽_{p})), we have lim_{s} M^{n} = 0 and Rlim_{s} M^{n} = 0. The first short exact sequence shows that N^{n} = 0 for all n, so that N is trivial. Then N^{n}(X) = 0, and the second shows that we have conditional convergence, so that Theorem 8.2 applies.

In order to apply this result, it is usually necessary to p-complete M first. If each M^{n} is finitely generated, the completed theory is simply M^{∗}(−; ℤ̂_{(p)}). Completion of M leaves the E_{1}-term and therefore the whole spectral sequence undisturbed, by Proposition 4.11.

15 Adams spectral sequences

We take M to be a ring spectrum, with multiplication ϕ: M ∧ M → M and unit η: S → M. Then M^{∗}(−) is a multiplicative cohomology theory. Classically, M = H(𝔽_{p}) for some prime p (and this may still be the most important case). We work stably, so that all cohomology groups are taken in the reduced sense.

An Adams tower for a spectrum Y is a diagram of spectra of the form

[figure: …; Y^{s+1}; Y^{s}; …; Y^{2}; Y^{1}; Y^{0} = Y; …; K^{s}; …; …; K^{1}; K^{0}; i; j; i; j; i; j; k; k; k]  (15.1)

in which each triangle is an exact triangle and certain axioms hold (which are subject to some negotiation). We extend the diagram to the right by taking Y^{s} = Y and K^{s} trivial for all s < 0. We obtain an unrolled exact couple simply by mapping any spectrum X into diagram (15.1),

A^{s} = {X, Y^{s}}^{∗},  E^{s} = {X, K^{s}}^{∗}.  (15.2)

This results in a half-plane spectral sequence with entering differentials, known as the Adams spectral sequence (or Adams–Novikov spectral sequence if M = MU or BP). Its target group is clearly A^{−∞} = A^{0} = {X, Y }^{∗}.

Following Adams [1, p. 316], the direct way to construct an Adams tower is to start from the exact triangle

S ⟶^{η} M ⟶^{q} \overline{M} ⟶^{i} S

formed from η, in which we arrange deg(i) = 0. We define inductively K^{s} = M ∧ Y^{s} and Y^{s+1} = \overline{M} ∧ Y^{s}, with maps formed by smashing this exact triangle with Y^{s}. Thus Y^{s} = \overline{M} ∧ … ∧ \overline{M} ∧ Y and K^{s} = M ∧ \overline{M} ∧ … ∧ \overline{M} ∧ Y, each with s copies of \overline{M}. The obvious splitting of j^{∗}: M^{∗}(K^{s}) = M^{∗}(M ∧ Y^{s}) → M^{∗}(Y^{s}), given by lifting x ∈ M^{∗}(Y^{s}) = {Y^{s}, M}^{∗} to ϕ^{∘}(M ∧ x), implies the long exact sequence

… ⟶ M^{∗}(K_{2}) ⟶ M^{∗}(K_{1}) ⟶ M^{∗}(K_{0}) ⟶ M^{∗}(Y) ⟶ 0,  (15.3)

which is in some sense a resolution of M^{∗}(Y). This leads to a description of the E_{2}-term (which we do not pursue in this generality).

We are primarily interested in convergence. There is no reason why the homotopy limit Y^{∞} = holim_{s} Y^{s} should be trivial, but we can use Theorem 4.9 to remove it. We define spectra Z^{s} by the exact triangles

Y^{∞} ⟶^{ϵ^{s}} Y^{s} ⟶ Z^{s} ⟶ Y^{∞},

as in (4.8), to replace the given Adams tower (15.1) by the “quotient” tower

[figure: …; Z^{s+1}; Z^{s}; …; Z^{2}; Z^{1}; Z^{0}; …; K^{s}; …; …; K^{1}; K^{0}]  (15.4)

where holim_{s} Z^{s} is trivial by Theorem 4.9(b). The new unrolled exact couple has

A^{s} = {X, Z^{s}}^{∗},  E^{s} = {X, K^{s}}^{∗} for s ≥ 0,  (15.5)

with A^{∞} = RA^{∞} = 0, as desired. Moreover, the obvious morphism of towers from (15.1) to (15.4) induces an isomorphism of spectral sequences, since the E_{1}-terms are the same; therefore the Adams spectral sequence converges conditionally to the revised colimit target group A^{−∞} = A^{0} = {X, Z^{0}}^{∗}. Of course, the target group is unchanged if Y^{∞} happens to be trivial, or more generally, if {X, Y^{∞}}^{∗} = 0; the latter occurs often enough to be useful (see Proposition 4.11). (We do not claim that (15.4) is an Adams tower for Z^{0}; this may or may not be true, but either way, is irrelevant.)

To complete the discussion, we have only to identify Z^{0}.

The classical case Take M = H(𝔽_{p}), and denote by A the mod p Steenrod algebra of operations on H^{∗}(−; 𝔽_{p}), which we identify with H^{∗}(H(𝔽_{p}); 𝔽_{p}).

Theorem 15.6 Let Y be a spectrum of finite type and X any spectrum. Then the Adams spectral sequence

E_{2}^{s,∗} = Ext_{A}^{s,∗}(H^{∗}(Y; 𝔽_{p}), H^{∗}(X; 𝔽_{p})) ⟹ {X, Y ∧ Ŝ_{(p)}}^{∗}

is conditionally convergent in the colimit sense. If, further, H^{n}(X; 𝔽_{p}) is finite for all n and zero for sufficiently large n, we have strong convergence.

Proof The recognition of the E_{2}-term is standard. For such Y, diagram (15.3) is indeed a resolution of H^{∗}(Y; 𝔽_{p}) by free A-modules and, because each K^{s} is of finite type, we have isomorphisms

E_{1}^{s} = {X, K^{s}}^{∗} ≅ Hom_{A}^{∗}(H^{∗}(K^{s}; 𝔽_{p}), H^{∗}(X; 𝔽_{p}))

that make the homological algebra work. The extra hypothesis on X forces each E_{2}^{s,t} to be finite, so that RE_{∞} vanishes trivially and Theorem 7.1 applies. For the target group, see Theorem 15.9(b), below.

We give one application. The result is due to Margolis [10], but we follow T. Y. Lin’s approach [8].

Theorem 15.7 (Margolis) For any spectrum Y of finite type, the cohomology functor H^{∗}(−; 𝔽_{p}) induces an isomorphism

{H(𝔽_{p}), Y }^{∗} ⟶^{≅} Hom_{A}^{∗}(H^{∗}(Y; 𝔽_{p}), A).

Proof We take X = H(𝔽_{p}) in Theorem 15.6. By Proposition 4.11, completion is not needed here. The key fact (Adams–Margolis [2]) is that A is an injective A-module, so that E_{2}^{s} = 0 for all s > 0. The spectral sequence collapses and we trivially have RE_{∞} = 0, hence strong convergence by Theorem 7.1. Therefore the edge homomorphism must be an isomorphism.

Remark Examples show that this result fails without some restriction on Y. We needed the finite type assumption in order to obtain the stated E_{2}-term.

We deduce a theorem of Lin [8, Thm. 3.2].

Corollary 15.8 For the Eilenberg–MacLane spectrum H(𝔽_{p}):

(a) The cohomotopy groups π^{∗}(H(𝔽_{p})) vanish;

(b) The Spanier–Whitehead dual DH(𝔽_{p}) of H(𝔽_{p}) is trivial.

Proof We take Y = S in Theorem 15.7. It is easy to see directly that Hom_{A}^{∗}(𝔽_{p}, A) = 0, which gives (a). By definition, π_{∗}(DH(𝔽_{p})) = π^{∗}(H(𝔽_{p})) = 0, which gives (b).

The general case There are many ring spectra M for which we can identify the target of the Adams spectral sequence, following Bousfield [3].

Theorem 15.9 Suppose the ring spectrum M is (−1)-connected, and consider the Adams spectral sequence arising from the tower (15.1).

(a) If the ring π_{0}(M) = ℤ_{J} for the set of primes J, then the Adams spectral sequence converges conditionally to the colimit {X, Y_{J}}^{∗}, where Y_{J} is the J-localization of Y.

(b) If the ring π_{0}(M) = ℤ/n, then the Adams spectral sequence converges conditionally to the colimit {X, Ŷ_{J}}^{∗}, where J is the set of primes dividing n and Ŷ_{J} is the J-completion of Y. If Y is of finite type, then Ŷ_{J} = Y ∧ Ŝ_{J}.

Proof Bousfield’s Theorems 6.5 and 6.6 in [3] identify Z^{0} in these cases as the Mhomology localization Y_{M} of Y, which we described above.

Remark The use of conditional convergence allows us to dispense entirely with conditions on X. (We do not compute the E_{2}-term.) Bousfield’s theorems are actually stated for more general M, when the core of the ring π_{0}(M) is ℤ_{J} or ℤ/n. This disposes of two of the four possible cases of the core subring; the other two admit no such result, according to Bousfield [3, Thm. 6.7].

References 39

References

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[2] J. F. Adams, H. R. Margolis, Modules over the Steenrod algebra, Topology 10 (1971), 271–282.

[3] A. K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), 257–281.

[4] H. Cartan, S. Eilenberg, Homological algebra, Princeton Univ. Press (1956).

[5] S. Eilenberg, J. C. Moore, Limits and spectral sequences, Topology 1 (1962), 1–23.

[6] D. M. Kan, Semisimplicial spectra, Illinois J. Math. 7 (1963), 463–478.

[7] A. G. Kurosh, The theory of groups, Chelsea (New York, 1956).

[8] T. Y. Lin, Duality and Eilenberg–MacLane spectra, Proc. Amer. Math. Soc. 56 (1976), 291– 299.

[9] S. MacLane, Homology, Grundlehren Math. Wiss. 114, Springer-Verlag (Berlin, 1963).

[10] H. R. Margolis, Eilenberg–MacLane spectra, Proc. Amer. Math. Soc. 43 (1974), 409–415.

[11] J. McCleary, User’s Guide to Spectral Sequences, 2nd Edition, Cambridge Univ. Press (New York, 1998).

[12] J. Milnor, Axiomatic homology theory, Pacific J. Math. 12 (1962), 337–341.

[13] E. H. Spanier, Algebraic topology, McGraw-Hill (New York, 1966).

[14] R. M. Vogt, Homotopy limits and colimits, Math. Z. 134 (1973), 11–52.

[15] C. T. C. Wall, Resolutions for extensions of groups, Math. Proc. Cambridge Philos. Soc. 57 (1961), 251–255.

Department of Mathematics Johns Hopkins University Baltimore, Maryland, U.S.A.

                                        E-mail: boardman@math.jhu.edu