ExampleConstruction: AI-generatedVerification: 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.

The right derived functors of Hom from a fixed object

Example

Assume the Axiom of Dependent Choice. Fix an object $X$ in an abelian category $A$ , and consider the covariant functor $A (X, -) : A \to A b$ . For supplied injective resolution data $I$ on a class $D$ , $R_{I}^{0} A (X, -) ≅ A (X, -),$ and if $J \in D$ is injective then $R_{I}^{n} A (X, -) (J) = 0 (n > 0) .$ This is the basic right-derived pattern that later becomes Ext.

Facts & Assumptions

Given: The Axiom of Dependent Choice, an object $X$ , supplied injective resolution data $I$ , and an injective object $J \in D$ .

[L1]

Hom is left exact in each variable, so $A (X, -)$ is left exact (Hom is left exact in each variable).

[L2]

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

[L3]

Positive right derived functors vanish on injective objects (Positive right derived functors vanish on injective objects).

Verification

technique · direct

1.1L1L2given

By [L1], the functor $A (X, -)$ satisfies the hypothesis of [L2], so $R_{I}^{0} A (X, -) ≅ A (X, -)$ .

2.1L3step 1.1∎

If $J$ is injective, then [L3] gives $R_{I}^{n} A (X, -) (J) = 0$ for every $n > 0$ . This is exactly the displayed example.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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

Joseph J. Rotman, An Introduction to Homological Algebra, 2nd ed. (standard reference, not scraped)