LemmaStatement: AI-generatedProof: AI-generated

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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.

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Graded pieces of a direct-sum tail tower

Statement

Let $(G_{j})_{j \geq 0}$ be spectra and put $T^{q} = ⨁_{j \geq q} G_{j}$ for $q \geq 0$ , with $T^{q} = T^{0}$ for $q \leq 0$ . Give these spectra the canonical tail inclusions and augmentation to $T^{0}$ . Then ${gr}^{q} T ≃ G_{q}$ for $q \geq 0$ , the connecting map $G_{q} \to T^{q + 1} [1]$ is zero, and ${gr}^{q} T ≃ 0$ for $q < 0$ . The identifications are natural in maps of families $(G_{j})$ .

Facts & Assumptions

Given: The family of spectra and its augmented tail tower.

Goal: Compute its graded pieces and connecting maps.

[F1]

Decreasing spectral filtrations and associated graded defines the associated graded as a cofiber.

[F2]

Cofiber of a direct-summand inclusion computes the cofiber and connecting map of a summand inclusion.

Proof

1.1F1F2givenconstruct

For $q \geq 0$ , the universal property of coproducts gives a natural equivalence $T^{q} ≃ T^{q + 1} \oplus G_{q}$ carrying the transition map to the first-summand inclusion. Applying [F2] gives the stated graded piece and zero connecting map.

2.1F1givenstep 1.1∎

For $q < 0$ the transition map is the identity of $T^{0}$ , whose cofiber is zero. The coproduct decompositions, inclusions and projections commute with maps of families, so the identifications are natural.

Depends on

Used by · 3

Dependency tree · 3 levels

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