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Short exact sequences of filtered complexes give compatible exact couples

Statement

Let $0 \to A \overset{α}{\to} B \overset{β}{\to} C \to 0$ be a short exact sequence of filtered complexes in an abelian category, whose maps are strict degreewise. Thus, identifying $A_{n}$ with its image in $B_{n}$ , one has $F_{p} A_{n} = A_{n} \cap F_{p} B_{n}$ and $β_{n} (F_{p} B_{n}) = F_{p} C_{n}$ , so $0 ⟶ F_{p} A_{n} ⟶ F_{p} B_{n} ⟶ F_{p} C_{n} ⟶ 0$ is exact for all $p, n$ . Then the associated-graded sequences are short exact sequences of complexes. The filtered maps induce morphisms between the three associated exact couples, commuting with $i, j, k$ , and hence compatible morphisms of all their derived couples and spectral sequences. This does not assert short exactness of the homology $D$ or $E$ terms.

Facts & Assumptions

[F1]

Filtered chain map preserves all filtration subcomplexes. A filtered complex produces an exact couple constructs $D_{p, q} = H_{p + q} (F_{p})$ and $E_{p, q} = H_{p + q} (F_{p} / F_{p - 1})$ with inclusion, quotient and connecting maps.

[F2]

Spectral sequence subquotient and local lifting calculus supplies epic local lifts, nested quotients and descent of subobject containments.

[F3]

Naturality of the homology connecting morphism gives the connecting square for a morphism of short exact sequences of complexes. Homology respects identities and composition preserves commuting chain-map squares under homology.

[F4]

A map of exact couples induces a map of spectral sequences derives and iterates commuting couple maps.

Proof

Given: The strict filtered short exact sequence. Fix $p, n$ and write ${gr}_{p} X = F_{p} X / F_{p - 1} X$ .

1.1

The restriction of $α_{n}$ to $F_{p} A_{n}$ is monic. Its image is $A_{n} \cap F_{p} B_{n}$ , exactly the kernel of the restriction of $β_{n}$ to $F_{p} B_{n}$ . Strict surjectivity makes the latter map epic onto $F_{p} C_{n}$ . The maps commute with the restricted differentials by [F1], so these are short exact sequences of subcomplexes for every $p$ .

F1F2

1.2

For either filtered map $f = α$ or $β$ , there is a commutative ladder from $0 \to F_{p - 1} X \to F_{p} X \to {gr}_{p} X \to 0$ to the corresponding sequence for its target $Y$ . Passing to homology gives the couple's $D$ and $E$ comparison maps. The squares for $i$ and $j$ commute because their chain maps are respectively filtration inclusions and quotient projections and homology preserves compositions. The square for $k : E_{p, q} \to D_{p - 1, q}$ is exactly the connecting square in [F3], with homology degree decreasing from $p + q$ to $p + q - 1$ . Hence all three couple squares commute at their prescribed bidegrees.

F1F3

2.1

The induced graded map from $A$ is monic: an element of $F_{p} A_{n}$ mapping into $F_{p - 1} B_{n}$ lies in $A_{n} \cap F_{p - 1} B_{n} = F_{p - 1} A_{n}$ . The graded map to $C$ is epic by lifting from $F_{p} C_{n}$ to $F_{p} B_{n}$ locally. If $b \in F_{p} B_{n}$ maps into $F_{p - 1} C_{n}$ , lift that image locally to $b^{'} \in F_{p - 1} B_{n}$ . Then $b - b^{'} \in \ker β_{n} \cap F_{p} B_{n}$ is the image of an element of $F_{p} A_{n}$ , and has the same graded class as $b$ . Conversely a graded class from $A$ maps to zero because $β α = 0$ . This proves both kernel-image containments. The element notation means morphisms after finite epic pullbacks, and all equalities descend by [F2]. Thus the graded sequence is short exact degreewise; its maps commute with differentials by quotient descent.

F1F2step 1.1

3.1

Apply [F4] to both maps from step 1.2 to obtain the derived-couple and spectral maps on all pages. The graded short exactness in step 2.1 is a statement at the chain level; step 1.2 applies homology and yields its natural connecting ladders, not a claim that each induced homology arrow is monic or epic. Zero complexes, equal successive filtration pieces and all integer indices are permitted throughout. Only finite epic lifting and canonical quotient arrows were used, without AC or a choice of splitting.

F4step 2.1step 1.2∎

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Sources

Weibel, An Introduction to Homological Algebra, Chapter 5 (standard reference, not scraped)
The Stacks Project, Homological Algebra (standard reference, not scraped)