PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 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.

Positive left derived functors vanish on projective objects

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. If $Q \in D$ is a projective object, then for every $n > 0$ , $L_{n}^{P} F (Q) = 0.$

Facts & Assumptions

Given: A projective object $Q \in D$ and an integer $n > 0$ .

[L1]

Every projective object admits a length-zero projective resolution (A projective object has a length-zero projective resolution).

[L2]

Changing the supplied projective resolution datum changes the derived objects only by natural isomorphism (Two supplied projective resolution data define naturally isomorphic left derived functors).

[L3]

The left derived object is the homology of the deleted chosen resolution (Left derived objects relative to supplied projective resolution data).

Proof

technique · direct

1.1

By [L1], the object $Q$ has a projective resolution concentrated in degree $0$ . Its deleted complex therefore has only one nonzero term, namely $Q$ in degree $0$ .

L1givenconstruct

2.1

Let $P^{'}$ be the supplied projective resolution datum on the same domain as $P$ that agrees with $P$ away from $Q$ and assigns the length-zero resolution from step 1.1 to $Q$ . By [L2], the derived object computed from $P$ is isomorphic to the one computed from $P^{'}$ . By [L3], the deleted resolution in $P^{'} (Q)$ is the one-term complex from step 1.1, whose homology is zero in every positive degree. Hence $L_{n}^{P} F (Q) = 0$ for $n > 0$ .

L2L3step 1.1∎

Depends on

Used by

An acyclic resolution that is not an injective resolution Example

Dependency tree · two levels

14 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

Romyar Sharifi, Homological Algebra (standard reference, not scraped)