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CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06
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Finite projective dimension is the largest nonzero Ext degree

Statement

If M≠0 has finite projective dimension d, then d=max⁡{k≥0:Ext⁡k(M,X)≠0 for some X}.

Facts & Assumptions

Given: A nonzero object M of finite projective dimension d under the stated enough-projectives and smallness hypotheses.

Proof

technique · direct
1.1givenconstruct

Projective dimension at most n iff higher Ext vanishes gives Ext⁡k(M,−)=0 for k>d.

2.1step 1.1algebra∎

If d=0, take X=M: then Ext⁡0(M,M)=Hom⁡(M,M) contains the nonzero identity of M. If d≥1 and every Ext⁡d(M,X) vanished, the same theorem with n=d−1 would give pd⁡(M)≤d−1, contradicting minimality of d. Hence in either case some X witnesses nonvanishing in degree d.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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