TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Injective dimension at most n iff higher Ext vanishes

Statement

Assume enough injectives. For an object $N$ and $n \geq 0$ , $id (N) \leq n$ if and only if ${Ext}^{k} (M, N) = 0$ for every object $M$ and every $k > n$ .

Facts & Assumptions

Given: An object $N$ , an integer $n \geq 0$ , and enough injectives.

Proof

technique · direct

1.1givenconstruct

If $id (N) \leq n$ , the $n$ th cosyzygy is injective. Repeated second-variable shifting from Ext dimension shifting in the second variable reduces every ${Ext}^{k} (M, N)$ with $k > n$ to positive Ext into that injective cosyzygy, which vanishes by Positive injective-resolution Ext vanishes on an injective second variable.

2.1step 1.1algebra∎

Conversely, repeated shifting makes ${Ext}^{1} (M, Σ^{n} N)$ vanish for every $M$ . The defining injective-dimension convention in Injective dimension of an object and the splitting criterion for the final cosyzygy show it is injective, so the chosen resolution has length at most $n$ .

Depends on

Used by

Dependency tree · two levels

9 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, Chapters 3–4 (standard reference, not scraped)