TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)

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Five term exact sequence of a first quadrant homological spectral sequence

Statement

Let $E_{p, q}^{r}$ be a homological spectral sequence in an abelian category, first quadrant from page $2$ , with specified finite abutment to $H_{n}$ and increasing filtration normalized by $F_{- 1} H_{n} = 0$ , $F_{n} H_{n} = H_{n}$ for $n \geq 0$ . There is an exact sequence $H_{2} ⟶ E_{2, 0}^{2} \overset{d^{2}}{\to} E_{0, 1}^{2} ⟶ H_{1} ⟶ E_{1, 0}^{2} ⟶ 0.$ The maps adjacent to homology are the edge maps. No initial injectivity of $H_{2} \to E_{2, 0}^{2}$ is asserted.

Facts & Assumptions

[F1]

Homological spectral sequence gives degree $(- r, r - 1)$ and homology transitions.

[F2]

Edge homomorphisms of a first quadrant spectral sequence specifies the homological edge maps through normalized extreme filtration pieces.

[F3]

Spectral sequence subquotient and local lifting calculus gives canonical kernel, image and quotient comparisons.

Proof

Given: The sequence and normalized abutment in the statement. All page indices below satisfy $r \geq 2$ ; terms outside the first quadrant stay zero.

1.1

At $(1, 0)$ the outgoing target $(1 - r, r - 1)$ and incoming source $(1 + r, 1 - r)$ are zero. Thus $E_{1, 0}^{2}$ is canonically $E_{1, 0}^{\infty} = H_{1} / F_{0} H_{1}$ , and the edge $H_{1} \to E_{1, 0}^{2}$ is the epic quotient map.

F1F2

1.2

At $(0, 1)$ all outgoing targets $(- r, r)$ vanish. Its incoming source is $(r, 2 - r)$ , which lies in the quadrant only for $r = 2$ . Therefore $E_{0, 1}^{\infty} = coker (d^{2} : E_{2, 0}^{2} \to E_{0, 1}^{2})$ , identified by abutment with $F_{0} H_{1} / F_{- 1} H_{1} = F_{0} H_{1}$ . The edge $E_{0, 1}^{2} \to H_{1}$ is the cokernel projection followed by the filtration inclusion.

F1F2F3

1.3

At $(2, 0)$ all incoming sources $(2 + r, 1 - r)$ vanish. Its outgoing target $(2 - r, r - 1)$ lies in the quadrant only at $r = 2$ . Hence $E_{2, 0}^{\infty} = \ker d^{2}$ , identified with $F_{2} H_{2} / F_{1} H_{2} = H_{2} / F_{1} H_{2}$ . The edge $H_{2} \to E_{2, 0}^{2}$ is the quotient onto that kernel followed by its monic inclusion.

F1F2F3

2.1

Step 1.3 proves that the image at $E_{2, 0}^{2}$ is $\ker d^{2}$ . Step 1.2 proves that the next kernel at $E_{0, 1}^{2}$ is $im d^{2}$ , and its image in $H_{1}$ is $F_{0} H_{1}$ . This is the kernel of the quotient in step 1.1, which is onto, proving exactness also at $E_{1, 0}^{2}$ before the terminal zero. These are precisely all claimed positions. There is no claim about a kernel at the initial $H_{2}$ without a preceding map. Zero terms and zero $d^{2}$ give the same kernel and cokernel factorizations; the degree-one normalized endpoints are used explicitly. All maps are canonical from the specified data, without splittings or AC.

F3step 1.1step 1.2step 1.3∎

Used by

Nothing in the library uses this result yet.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

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