LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

global dimension is detected on cyclic modules

Statement

For a unital ring $R$ , its left global dimension equals $\sup_{I} {pd}_{R} (R / I)$ over all left ideals $I$ , and equals the supremum of the injective dimensions of all left modules. The equalities allow infinity; in the commutative Noetherian case the cyclic modules are finite.

Facts & Assumptions

Given: The objects and hypotheses in the statement. We work with the Axiom of Choice; cited dependent-choice and resolution-existence hypotheses are retained.

[F1]

Left and right global dimension of a ring: For a ring $R$ , define $l.gl.dim R = \sup {{pd}_{R} M : M is a left R -module},$ and $r.gl.dim R = \sup {{pd}_{R^{o p}} M : M is a right R -module} .$ They are separately defined extended natural numbers; their equality is not part of the notation.

[F2]

Baer's criterion for injective modules: Assume the Axiom of Choice. A left $R$ -module $I$ is injective if and only if every homomorphism $f : J \to I$ from a left ideal $J \leq R$ extends to a homomorphism $R \to I$ . The forward implication is choice-free. The converse uses AC through Zorn's lemma.

[F3]

Injective dimension at most n iff higher Ext vanishes: 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 $k > n$ .

[F4]

Projective dimension at most n iff higher Ext vanishes: Assume the Axiom of Dependent Choice. In an abelian category with enough projectives and enough injectives, fix supplied projective and injective resolution data on all objects. Let $M$ be an object and $n \geq 0$ . The following are equivalent: 1. $pd (M) \leq n$ ; 2. ${Ext}^{k} (M, N) = 0$ for every object $N$ and every $k > n$ ; 3. ${Ext}^{n + 1} (M, N) = 0$ for every object $N$ .

[F5]

Ext dimension shifting in the second variable: Assume the Axiom of Dependent Choice. Let $A$ be abelian with enough projectives and enough injectives, and fix supplied projective and injective resolution data on all its objects. If $0 \to N \to I \to Σ N \to 0$ is an injective copresentation, then for $q \geq 1$ there are natural isomorphisms ${Ext}^{q + 1} (M, N) ≅ {Ext}^{q} (M, Σ N)$ ; its low-degree part is $0 \to Hom (M, N) \to Hom (M, I) \to Hom (M, Σ N) \to {Ext}^{1} (M, N) \to 0$ .

[F6]

Module categories have enough injectives: Assume the Axiom of Choice. For every unital ring $R$ and every left $R$ -module $M$ , there is an injective left $R$ -module $I$ and a monomorphism $M \to I$ . Thus left $R$ -modules have enough injectives. For commutative $R$ , one explicit functorial target is $J (M) = {(R^{(M^{\lor})})}^{\lor} ≅ \prod_{ϕ \in M^{\lor}} R^{\lor},$ where $X^{\lor} = {Hom}_{Z} (X, Q / Z)$ ; the embedding is $M \to M^{\lor \lor} \to J (M)$ . Here $X^{\lor}$ is a left $R$ -module by $(r ϕ) (x) = ϕ (r x)$ .

Proof

1.1

Fix $n \geq 0$ and suppose every $R / I$ has projective dimension at most $n$ . For any left module $N$ , choose an injective resolution and let $C$ be its $n$ th cosyzygy, with $C = N$ when $n = 0$ . Dimension shifting gives ${Ext}^{1} (R / I, C) = {Ext}^{n + 1} (R / I, N) = 0$ . The last vanishing follows from the projective-dimension Ext criterion.

F6F5F4

2.1

To apply Baer, any map $I \to C$ extends to $R$ : its pushout with $I ↪ R$ yields an extension of $R / I$ by $C$ , whose Ext class is zero and hence splits. Therefore $C$ is injective by Baer. The truncated injective resolution gives $id N \leq n$ , so all ${Ext}^{j} (M, N)$ vanish for $j > n$ and arbitrary $M, N$ .

F2F3step 1.1

3.1

The projective-dimension criterion now gives $pd M \leq n$ for every module $M$ . Conversely such a global bound applies to all cyclic modules and forces every injective dimension at most $n$ by the same Ext criterion. Thus all three bounds are equivalent for each finite $n$ , proving equality of their extended suprema. This includes the zero ring, whose only module has dimension zero under the adopted resolution convention.

F4F3F1step 2.1∎

Depends on

Used by

Dependency tree · two levels

26 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

Propositions 12.24–12.25, p.120 (standard reference, not scraped)