Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06
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Global dimension is the supremum of nonzero Ext degrees

Statement

For an abelian category with enough projectives, gldim⁡A=sup⁡{n:Ext⁡n(M,N)≠0 for some M,N}; the supremum is taken in N0∪{∞}, with sup⁡∅=0. If A also has enough injectives, the dual supremum of injective dimensions has the same value.

Facts & Assumptions

Given: The stated enough-projectives or enough-injectives hypothesis.

Proof

technique · direct
1.1givenconstruct

By Global dimension of an abelian category, global dimension is the supremum of projective dimensions. Projective dimension at most n iff higher Ext vanishes says that each individual projective dimension is exactly the least uniform Ext-vanishing bound.

2.1step 1.1algebra∎

Taking suprema over objects gives the displayed equality. If there is no nonzero object, both sides are 0 by the stated convention; otherwise Ext⁡0(M,M) contains 1M≠0 for every nonzero M, so the degree-zero endpoint is present. When there are also enough injectives, Injective dimension at most n iff higher Ext vanishes gives the same uniform Ext bounds and hence the dual formulation.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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