TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30

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.

Buchsbaum-Eisenbud rank and grade criterion for exact free complexes

Statement

Assume the Axiom of Choice. Let $(R, m)$ be a Noetherian local ring and $F_{∙} : 0 \to R^{n_{e}} \overset{d_{e}}{\to} \dots \overset{d_{1}}{\to} R^{n_{0}}$ a finite free complex. Define $r_{i} = n_{i} - n_{i + 1} + \dots + (- 1)^{e - i} n_{e}$ . When $r_{i} \geq 0$ , let $I_{i}$ be the ideal of $r_{i}$ -minors of $d_{i}$ , with the $0$ -minor ideal equal to $R$ . Then the complex is exact at every positive-degree term if and only if, for all $1 \leq i \leq e$ , the integers $r_{i}$ are nonnegative, all $(r_{i} + 1)$ -minors vanish, and either $I_{i} = R$ or $I_{i}$ contains an $R$ -regular sequence of length $i$ . Under these equivalent conditions, $d_{i}$ has rank exactly $r_{i}$ .

Facts & Assumptions

Given: The local Noetherian ring, finite free complex, expected ranks, and determinantal ideals.

[F1]

Positive-degree exactness forces expected rank, vanishing larger minors, and the regular-sequence alternative, including over nonreduced rings (Exact free complexes have the expected ranks and determinantal regular sequences).

[F2]

Those rank and regular-sequence conditions force positive-degree exactness (Expected ranks and determinantal regular sequences force a free complex to be exact).

Proof

technique · apply the separately proved necessity and sufficiency directions

1.1F1

If the complex is positively exact, [F1] gives nonnegative $r_{i}$ , vanishing $(r_{i} + 1)$ -minors, the determinantal regular-sequence alternative, and exact rank $r_{i}$ for every $i$ .

2.1F1F2step 1.1∎

Conversely, assume the conditions in the Statement. By [F2] the complex is exact in every positive degree, and [F2] also gives exact rank $r_{i}$ . AC is inherited by the two proved directions.

Depends on

Used by

Fibrewise exactness of a finite free complex is open in a flat Cohen-Macaulay family Lemma

Dependency tree · two levels

14 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, Algebra, Proposition 10.102.9 (tag 00N1), exactness criterion (standard reference, not scraped)