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.

Degreewise eventually zero spectral towers have zero limit

Statement

Assume the Axiom of Choice and the fixed compatible universe conventions of the stable spectra interface. Let $\dots \overset{f_{1}}{\to} E_{1} \overset{f_{0}}{\to} E_{0}$ be a coherent tower of spectra. Put $P = \prod_{r \geq 0} E_{r}$ , let $p_{r} : P \to E_{r}$ be the projection, and let $s : P \to P$ be the shift specified by $p_{r} s = f_{r} p_{r + 1}$ . Then $\lim_{r \geq 0} E_{r} ≃ fib (1 - s)$ and $π_{n} P ≅ \prod_{r \geq 0} π_{n} E_{r}$ for every integer $n$ . If for every $n$ there is $N_{n} \geq 0$ with $π_{n} E_{r} = 0$ whenever $r \geq N_{n}$ , then $\lim_{r \geq 0} E_{r} ≃ 0$ .

Facts & Assumptions

Given: Choice, the fixed universe conventions, and the small coherent tower in the statement; for the zero-limit conclusion, its stated degreewise eventual vanishing.

Goal: Apply the imported fiber formula, calculate the product groups, and prove the zero-limit consequence.

[A1]

Assume The Axiom of Choice, retaining the foundations of Stable spectra and exact triangles. It also permits simultaneous representatives and paths in the component calculation for products below.

[F1]

Stable spectra and exact triangles supplies the stable category, coherent small products and limits, the zero spectrum, and $π_{n} Z = π_{0} Map (S [n], Z)$ . Its homotopy-group detection criterion says that a spectrum with all integer homotopy groups zero is equivalent to zero.

[F2]

Homotopy exactness of a spectral fiber sequence supplies the long exact sequence for a fiber sequence of spectra, in all integer degrees.

[F3]

Reference 0 imports the sequential-limit fiber formula, applicable to the stable category in [F1] because it has countable products.

External input

Use Cnossen, Corollary 4.4.35, to identify the limit with the fiber of $1 - s : P \to P$ , with shift components $p_{r} s = f_{r} p_{r + 1}$ . The difference map and its fiber identification are imported from the cited stable-category formula. The product-group calculation and eventual-vanishing application are local.

Proof

1.1A1F1F3given

By [F1] and [F3], $P$ , its shift and the asserted fiber sequence exist. The universal property of products identifies $Map (S [n], P)$ with $\prod_{r \geq 0} Map (S [n], E_{r})$ . Components of this product are the product of the component sets: under [A1], a family of components admits a family of representatives, and representatives in the same component in every factor admit a family of paths, which gives a path in the product. Thus the projection map on components is bijective; it is a homomorphism by functoriality of spectral homotopy groups, so it is the asserted group isomorphism. Under it, $π_{n} (s)$ is the homomorphism $S_{n}$ with $(S_{n} x)_{r} = (f_{r})_{*} (x_{r + 1})$ .

2.1F1F2step 1.1givenalgebra∎

Under eventual vanishing, set $M_{n} = \max (1, N_{n})$ . The group $\prod_{r \geq 0} π_{n} E_{r}$ has zero factors for $r \geq M_{n}$ . Every coordinate of $S_{n}^{M_{n}} x$ comes from $x_{r + M_{n}} = 0$ , so $S_{n}^{M_{n}} = 0$ . Hence $1 - S_{n}$ has two-sided inverse $\sum_{j = 0}^{M_{n} - 1} S_{n}^{j}$ , by multiplying the finite sum and cancelling adjacent terms. The same holds in degree $n + 1$ . Apply [F2] to the fiber sequence of step 1.1: the relevant segment is $π_{n + 1} P \overset{1 - S_{n + 1}}{\to} π_{n + 1} P \to π_{n} \lim_{r} E_{r} \to π_{n} P \overset{1 - S_{n}}{\to} π_{n} P$ . The first map is onto and the last map injective, so exactness forces $π_{n} \lim_{r} E_{r} = 0$ . This holds for every integer $n$ , and the detection criterion of [F1] makes the limit zero.

Depends on

Used by · 1

Stage bounds imply completeness and finite multiplicative convergence

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

Bastiaan Cnossen, Stable Homotopy Theory and Higher Algebra · Section 4.4.6, Mapping telescopes and the Milnor sequence; Lemma 4.4.34 and Corollary 4.4.35 (Dual mapping telescope), online section last updated 6 November 2026

In a stable infinity-category with countable products, a coherent sequential inverse limit is the fiber of identity minus the transition shift on the product of the stages.

Hypotheses: A small coherent tower in a stable infinity-category admitting countable products. No connectivity or eventual-vanishing hypothesis is needed for this formula.

Reading limits: Freshly read the author’s online Section 4.4.6 at Lemma 4.4.34 and Corollary 4.4.35, including their proofs. The PDF was unavailable through the web tool; no fresh PDF or page-image reading is claimed. Import the categorical fiber formula and its specified difference map. Apply it to the supplied category of spectra; prove the product-group and eventual-vanishing consequences locally. Earlier draft reading claims grant no fresh credit, and upstream categorical foundations are not reconstructed.

Sources

Bastiaan Cnossen, Stable Homotopy Theory and Higher Algebra · Section 4.4.6, Mapping telescopes and the Milnor sequence; Lemma 4.4.34 and Corollary 4.4.35 (Dual mapping telescope), online section last updated 6 November 2026