LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12

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The filtered differential induces d r on the r page

Statement

For each $r \geq 0$ , the filtered differential induces a morphism $d^{r} : E_{p, q}^{r} ⟶ E_{p - r, q + r - 1}^{r}$ characterized on local representatives by $[x] \mapsto [d_{n} x]$ , where $n = p + q$ .

Facts & Assumptions

Given: A filtered complex, $n = p + q$ , and $r \geq 0$ ; symbols $A^{r}, B^{r}$ have [F1] conventions.

[F1]

$E^{r}$ is the quotient of the specified $A^{r}$ numerator by its two denominator summands (R page of the spectral sequence of a filtered complex).

[F2]

The associated graded differential is well defined (The associated graded of a filtered complex is a bigraded complex).

[F3]

Maps preserving numerator and denominator descend uniquely; image representatives can be lifted after epic pullback (Spectral sequence subquotient and local lifting calculus).

Proof

technique · direct

1.1F1F2

For r=0 use [F2]. For r≥1, the restriction of d to $A_{p, n}^{r}$ lands in $F_{p - r} C_{n - 1}$ . Its next differential is zero, so it lands in $A_{p - r, n - 1}^{r}$ , the target numerator in [F1].

1.2F1F3algebra

If $x$ changes by $a \in A_{p - 1, n}^{r - 1}$ , then $d a$ lies in the target boundary summand $d (A_{(p - r) + r - 1, n}^{r - 1}) = d (A_{p - 1, n}^{r - 1})$ . If x changes by $d b$ with $b \in A_{p + r - 1, n + 1}^{r - 1}$ , its differential changes by $d^{2} b = 0$ . These are subobject factorization calculations: the sum epimorphism allows their verification after an epic cover by [F3].

2.1F3step 1.1step 1.2∎

Consequently the map from the source numerator to the target quotient kills both denominator summands and hence their sum. Cokernel descent [F3] gives a unique $d^{r}$ satisfying the stated rule. Equality after the epic source quotient proves that the rule is independent of every permitted local representative.

Source notes

Weibel, Chapter 5, Construction 5.4.6 and Lemma 5.4.7, pp.133–134; Sharifi, Theorem 4.2.3, pp.91–92. Increasing homological indices are used here.

Depends on

Used by

An exhaustive nonseparated filtration with the wrong naive abutment Counterexample
Collapse with a nonsplit extension problem Counterexample
Identical E2 pages with different later differentials Counterexample
A filtered complex collapsing at e one Example
A first quadrant five term exact sequence Example
A nonzero d two differential in a small filtered complex Example
Edge maps from a first quadrant spectral sequence Example
The cohomological reindexing of a homological spectral sequence Example
The two step filtration and its spectral sequence Example
A spectral sequence collapses when one differential is zero False statement
An isomorphism on e infinity automatically gives an isomorphism of unfiltered targets False statement
Collapse solves all extension problems False statement
Multiplicative filtered cochains induce products on every spectral-sequence page Lemma
The rth differential squares to zero Lemma
An E2 page alone does not determine the abutment Proposition
Collapse does not in general split the abutment Proposition
E one is homology of the associated graded complex Proposition
E zero is the associated graded complex Proposition
Failure of separatedness or completeness can destroy the claimed abutment Proposition
Serre transgression agrees with the relative connecting construction Proposition
The exact couple and subquotient constructions of the filtered complex spectral sequence agree Proposition
The rth differential has bidegree minus r r minus one Proposition
Complete exhaustive filtered complex convergence criterion Theorem
Homological Serre spectral sequence Theorem
The next page is the homology of the current page Theorem
The row filtration spectral sequence of a first quadrant double complex Theorem

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 5 (standard reference, not scraped)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)