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.

Filtered equivalences preserve image filtrations

PDF p. 4: Property (4): comparison of filtrations

Statement

Let $F^{∙}, G^{∙}$ be integer-indexed decreasing towers of spectra and $e : F^{∙} ≃ G^{∙}$ an equivalence compatible with their coherent transition maps. Identify $F^{0}$ and $G^{0}$ with the same spectrum $E$ so that $e^{0}$ becomes the identity. For every $j \geq 0$ and $i \in Z$ , the isomorphism $π_{i} (e^{j})$ identifies the images of the two transition maps to stage zero inside $π_{i} E$ .

More generally, suppose the towers carry compatible augmentations $a_{F}^{j} : F^{j} \to E$ and $a_{G}^{j} : G^{j} \to E$ , and $e$ respects them: $a_{G}^{j} e^{j} ≃ a_{F}^{j}$ for every integer $j$ . Then their augmentation images agree for every $i, j \in Z$ . If the stage-zero augmentations are the chosen identifications, these are the transition-to-zero images for $j \geq 0$ . A decreasing tower supplies a transition $j \to k$ when $j \geq k$ ; it supplies no transition from a negative stage to zero.

Statement references

Decreasing spectral filtrations and associated graded

Images of filtration stages

Facts & Assumptions

Given: The towers, their equivalence and stage-zero identifications; for the second assertion, the specified compatible augmentations.

Goal: Compare the two image subgroups in the stated ranges.

[F1]

An augmentation induces a homomorphism of abelian homotopy groups whose image is a subgroup. Transitions in a decreasing tower go from larger indices to smaller ones. Decreasing spectral filtrations and associated graded, Images of filtration stages

Proof

1.1F1given

For $j \geq 0$ , the transition $j \to 0$ exists. Compatibility of $e$ with this transition gives a commutative square on homotopy groups with left vertical isomorphism $π_{i} (e^{j})$ and right vertical identity on $π_{i} E$ . An element in the first image therefore lies in the second; conversely, lift its source along the inverse left isomorphism. The images agree.

2.1F1givenstep 1.1∎

Under the additional augmentation compatibility, the same commutative square exists with the augmentation maps for every integer $j$ . The identical element argument proves equality of those images. For $j \geq 0$ , compatibility of each augmentation with the transition to zero and its stipulated stage-zero identification identifies the augmentation map with that transition followed by identification. Thus this general form includes the first form whenever these extra data are supplied.

Depends on

Used by · 1

Smooth motivic comparison

Dependency tree · 3 levels

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