PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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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.

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Left derived maps preserve composition

Statement

Assume the Axiom of Dependent Choice.

Let $P$ be a supplied projective resolution datum on a class $D$ and $F : A \to B$ an additive functor between abelian categories. For composable morphisms $A \overset{u}{\to} B \overset{v}{\to} C$ with $A, B, C \in D$ and every $n \in Z$ , $L_{n}^{P} F (v u) = L_{n}^{P} F (v) \circ L_{n}^{P} F (u) .$

Facts & Assumptions

Given: Composable morphisms $A \overset{u}{\to} B \overset{v}{\to} C$ with $A, B, C \in D$ and an integer $n$ .

[L1]

Each derived map is induced from a comparison lift on the supplied resolutions (The left derived map relative to supplied resolution data).

[L2]

A comparison lift of a composite is homotopic to the composite of comparison lifts (Comparison maps respect composition up to homotopy).

[L3]

Chain-homotopic maps induce the same homology map (Chain-homotopic maps induce the same map on homology).

[L4]

Homology respects composition (Homology respects identities and composition).

Proof

technique · direct

1.1

Choose comparison lifts ${\tilde{u}}_{∙}$ of $u$ , ${\tilde{v}}_{∙}$ of $v$ , and ${\tilde{v u}}_{∙}$ of $v u$ as in [L1]. By [L2], ${\tilde{v u}}_{∙}$ is homotopic to ${\tilde{v}}_{∙} {\tilde{u}}_{∙}$ .

L1L2givenconstruct

2.1

After applying $F$ , [L3] makes the induced homology map of $F ({\tilde{v u}}_{∙})$ equal to that of $F ({\tilde{v}}_{∙} {\tilde{u}}_{∙})$ . By [L4], the latter equals the composite of the maps induced by $F ({\tilde{u}}_{∙})$ and $F ({\tilde{v}}_{∙})$ . Translating back through [L1] gives $L_{n}^{P} F (v u) = L_{n}^{P} F (v) \circ L_{n}^{P} F (u)$ .

L1L3L4step 1.1∎

Depends on

Used by

Left derived functors relative to supplied data are additive functors Theorem

Dependency tree · two levels

12 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

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 2 `Derived Functors` (standard reference, not scraped)