LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Images of filtration stages

PDF p. 3: Section 2, image filtration; PDF p. 8: Section 4, opening paragraph

Statement

Let $F$ be a coherently augmented decreasing filtration on a spectrum $E$ . For integers $i, j$ , the image of $π_{i} F^{j} \to π_{i} E$ is a subgroup of $π_{i} E$ , and the same homomorphism, regarded as a map to this image, is surjective.

Statement references

Decreasing spectral filtrations and associated graded

Facts & Assumptions

Given: A spectrum $E$ , its augmented decreasing filtration $F$ , and integers $i, j$ .

Goal: Establish the subgroup and surjectivity assertions.

[F1]

Decreasing spectral filtrations and associated graded supplies the structure map $F^{j} \to E$ and the induced homomorphism of abelian homotopy groups.

Proof

1.1F1given

Write $u$ for the homomorphism in [F1]. Its image consists precisely of the elements $u (x)$ with $x \in π_{i} F^{j}$ . It contains zero and is closed under sums and negatives because $u$ is a homomorphism.

2.1step 1.1∎

Every element of this image has a preimage by its definition. Thus $u$ , with codomain restricted to its image, is surjective. This proves both assertions.

Depends on

Decreasing spectral filtrations and associated graded

Used by · 5

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