ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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.

One dimension shift along an injective copresentation

Example

Assume the Axiom of Dependent Choice.

Let $I$ be supplied injective resolution data on a class $D$ of objects of $A$ , let $F : A \to B$ be additive and left exact, and let $0 \to A \to J \to C \to 0$ be a short exact sequence in $D$ with $J$ injective. Then for every $n > 1$ the connecting map gives an isomorphism $R_{I}^{n - 1} F (C) \tilde{\to} R_{I}^{n} F (A) .$

Facts & Assumptions

Given: A short exact sequence $0 \to A \to J \to C \to 0$ in $D$ with $J$ injective and an integer $n > 1$ .

[L1]

The right derived functors form a cohomological delta functor (Right derived functors form a cohomological delta functor).

[L2]

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

[L3]

Vanishing of the adjacent injective terms makes the connecting map an isomorphism (Dimension shift for a cohomological delta functor effaced in the middle).

Verification

technique · direct

1.1L1givenconstruct

By [L1], the given short exact sequence yields an exact segment $R_{I}^{n - 1} F (J) \to R_{I}^{n - 1} F (C) \overset{\partial^{n - 1}}{\to} R_{I}^{n} F (A) \to R_{I}^{n} F (J) .$

2.1L2L3step 1.1∎

Since $J \in D$ is injective and $n > 1$ , [L2] gives $R_{I}^{n - 1} F (J) = R_{I}^{n} F (J) = 0$ . Hence [L3] makes the connecting map in step 1.1 an isomorphism.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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

The Stacks Project, Section 12.12: Cohomological delta-functors (standard reference, not scraped)
Charles A. Weibel, An Introduction to Homological Algebra, Chapter 2 `Derived Functors` (standard reference, not scraped)