ExampleConstruction: AI-generatedVerification: AI-adaptedprecheck passaudited 2026-09-04

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 left derived functors of an exact functor

Example

Let $P$ be supplied projective resolution data on a class $D$ and let $F : A \to B$ be exact. Then for every object $A \in D$ , $L_{0}^{P} F (A) ≅ F (A) and L_{n}^{P} F (A) = 0 for n > 0.$ So the full left-derived profile of an exact functor is concentrated in degree $0$ .

Facts & Assumptions

Given: An exact functor $F$ and an object $A \in D$ .

[L1]

The zero-th left derived functor of a right exact functor recovers the original functor (The zero-th left derived functor of a right exact functor recovers the functor).

[L2]

An exact functor has vanishing positive derived functors (An exact functor has vanishing positive derived functors).

Verification

technique · direct

1.1L1given

Exact functors are in particular right exact, so [L1] gives $L_{0}^{P} F (A) ≅ F (A)$ .

2.1L2step 1.1∎

Since $F$ is exact, [L2] gives $L_{n}^{P} F (A) = 0$ for every $n > 0$ . Therefore the example has exactly the displayed degree-zero profile.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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)