PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Morphisms from a homotopically projective complex need no roof

Statement

For a K-projective complex $P$ and any complex $X$ , $Q : {Hom}_{K} (P, X) \to {Hom}_{D} (P, X)$ is bijective, under the standing localization size convention.

Facts & Assumptions

Given: For a K-projective complex $P$ and any complex $X$ , $Q : {Hom}_{K} (P, X) \to {Hom}_{D} (P, X)$ is bijective, under the standing localization size convention.

[F1]

K-projectivity annihilates Hom into all acyclic shifts (Homotopically projective bounded above complex).

[F2]

The cone of a quasi-isomorphism is acyclic (A chain map is a quasi-isomorphism exactly when its cone is acyclic).

[F3]

Both representable Hom sequences of a distinguished triangle are exact (Long exact Hom sequences of a distinguished triangle).

[F4]

In the derived localization every morphism is represented by a roof, and for parallel ordinary arrows $b, c : P \to X$ , equality $Q (b) = Q (c)$ holds exactly when $b v = c v$ after precomposition by some quasi-isomorphism $v : V \to P$ (Derived category of an abelian category, The calculus of fractions constructs the localization).

Proof

1.1

For a quasi-isomorphism $s : U \to V$ its cone $C$ is acyclic. The exact sequence ${Hom}_{K} (P, C [- 1]) \to {Hom}_{K} (P, U) \to {Hom}_{K} (P, V) \to {Hom}_{K} (P, C)$ has zero outer terms. Thus postcomposition by $s$ is bijective, including when $P$ or either Hom group is zero.

F1F2F3

2.1

Represent an arrow from $P$ by $P \overset{s}{\leftarrow} U \overset{f}{\to} X$ . The previous bijection supplies a unique $a : P \to U$ in $K$ with $s a = 1_{P}$ , so the roof equals $Q (f a)$ . If $Q (b) = Q (c)$ for $b, c : P \to X$ , equality detection gives $v : V \to P$ a quasi-isomorphism with $b v = c v$ ; take $a : P \to V$ with $v a = 1_{P}$ to get $b = c$ . This proves surjectivity and injectivity.

F4step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

13.19.3–13.19.8; W 10.4.8 for the equivalence (standard reference, not scraped)