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.

Reindexing commutes with shifts

PDF p. 4: Shift [2j] in equation (2.1)

Statement

Let $D_{n} = C^{- n}$ be the chain complex associated to a cochain complex $C$ . For an integer $k$ , define the chain shift by $D [k]_{n} = D_{n - k}$ and differential $(- 1)^{k} \partial_{n - k}$ . The chain complex associated to $C [k]$ is exactly $D [k]$ .

Facts & Assumptions

Given: $C$ , $D$ , and $k$ as stated.

Goal: Compare the shifted chain complexes.

[F1]

Reindexing cochains as chains specifies the cochain-to-chain reindexing.

[F2]

Derived category of an abelian category specifies the cochain shift.

Proof

1.1F1F2given

In degree $n$ , the chain group associated to $C [k]$ is $C [k]^{- n} = C^{- n + k} = D_{n - k}$ .

2.1F1F2step 1.1∎

Its differential is $(- 1)^{k} d_{C}^{- n + k}$ , which is precisely $(- 1)^{k} \partial_{n - k}$ . Thus the identification is an equality of chain complexes, natural in $C$ .

Depends on

Used by · 1

Complex shifts and spectral homotopy indices

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.