PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-08-31

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.

The cone construction commutes with shift up to the canonical sign isomorphism

Statement

For every chain map $f : C_{∙} \to D_{∙}$ , there is a canonical chain isomorphism $σ_{f} : Cone (f [1]) \tilde{\to} Cone (f) [1] .$

Facts & Assumptions

Given: A chain map $f : C_{∙} \to D_{∙}$ .

[L1]

The shift satisfies $C [1]_{n} = C_{n - 1}$ and $d^{C [1]} = - d^{C}$ (The shift of a chain complex).

[L2]

The shifted map satisfies $f [1]_{n} = f_{n - 1}$ (Shifted chain maps and shifted chain homotopies).

[L3]

The cone differential is $d (y, x) = (d (y) + f (x), - d (x))$ (The mapping cone of a chain map).

Proof

technique · direct

1.1

The two complexes have the same degree- $n$ object $Cone (f [1])_{n} = D_{n - 1} \oplus C_{n - 2} = Cone (f) [1]_{n} .$ Define $σ_{f, n} (y, x) : = (- y, x) .$

L1L2L3givenconstruct

2.1

By [L1], [L2], and [L3], the differential on $Cone (f [1])$ is $(- d^{D} (y) + f (x), d^{C} (x)),$ while the shifted differential on $Cone (f) [1]$ is $(- d^{D} (y) - f (x), d^{C} (x)) .$ The sign in $σ_{f}$ changes the first component exactly enough to intertwine these two formulas, so $σ_{f}$ is a chain isomorphism.

L1L2L3step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Romyar Sharifi, Homological Algebra (standard reference, not scraped)
Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)