False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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fs-the-derived-category-is-the-same-category-as-the-homotopy-category.md

Statement

For every abelian category, $Q : K (A) \to D (A)$ is an equivalence of categories.

Facts & Assumptions

Given: For every abelian category, $Q : K (A) \to D (A)$ is an equivalence of categories.

[F1]

A complex is zero in the derived category iff it is acyclic (A complex is zero in the derived category exactly when it is acyclic).

Refutation

1.1

In abelian groups take $X^{0} = Z$ , $X^{1} = Z$ , $X^{2} = Z / 2$ , with $d^{0} = 2$ , $d^{1}$ reduction modulo two, and other terms zero. The first map is monic, its image is the kernel of the second, and the second is epic. Thus $X$ is acyclic and $Q X = 0$ .

F1algebra

2.1

A contraction would satisfy $1_{X^{2}} = d^{1} h^{2}$ , giving a section $h^{2} : Z / 2 \to Z$ . Every such homomorphism is zero because $Z$ has no nonzero element killed by two. Thus $1_{X} \neq 0$ in $K$ , whereas $Q (1_{X}) = 0$ in $D$ . The functor is not faithful and cannot be an equivalence. This is a direct verification of the familiar acyclic-but-not-contractible obstruction.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

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