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.

Naturality of image filtrations

PDF p. 3: Section 2, functorial filtration

Statement

Let $F$ and $G$ be decreasing filtrations on spectra $E$ and $D$ . Suppose maps $F^{j} \to G^{j}$ and $E \to D$ form a coherent map of augmented towers: in particular each square with the maps to $E$ and $D$ commutes up to the specified homotopy. Then $π_{i} E \to π_{i} D$ carries the $j$ th image subgroup of $E$ into that of $D$ , for every $i, j \in Z$ . These restricted maps respect identity and composition whenever the original maps do.

Facts & Assumptions

Given: The two augmented towers and compatible maps just specified.

Goal: Establish induced functorial maps on their image subgroups.

[F1]

Decreasing spectral filtrations and associated graded supplies the induced homomorphisms on homotopy groups; homotopic maps induce the same homomorphism.

[F2]

The image subgroups are defined and are subgroups by Images of filtration stages.

Proof

1.1F1F2given

Take an element in the image subgroup for $F^{j}$ . Choose a preimage in $π_{i} F^{j}$ . The homotopy-commutative square sends its image in $π_{i} E$ to the image in $π_{i} D$ of the chosen preimage under $π_{i} F^{j} \to π_{i} G^{j}$ . Hence the element lands in the stated subgroup.

2.1step 1.1∎

The restricted map is the restriction of the given homomorphism on ambient homotopy groups. Therefore identity and composition hold on the image subgroups whenever they hold on the ambient groups. This proves the claim.

Depends on

Used by · 1

Motivic filtration and image groups

Dependency tree · 3 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.