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TheoremStatement: Literature-sourcedProof: Literature-sourced
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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.

External maximal-minor height bound for the component-scope example

PDF p. 15: Theorem6.7 second Bertini application

Statement

For a matrix M of size m×n over a Noetherian ring and proper ideal It(M) of its t-minors, ht⁡It(M)≤(m−t+1)(n−t+1). In particular a proper maximal-minor ideal for a 2×3 matrix has a prime over it of height at most two. This input is used only to establish the precise disconnected fixed-pair counterexample.

Source contract

Given: The stated Noetherian ring, matrix and proper determinantal ideal.

Goal: Import the exact height bound needed for that example.

[F1] Bruns–Vetter, Determinantal Rings, section2.A, Theorem(2.1), with Lemma(2.3) used in its printed proof.

External input

Import [F1], with its exact source record. No arbitrary prime-avoidance or expected-height assertion is substituted for this bound.

Depends on

Used by · 2

Dependency tree · 1 level

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.

External dependencies

Winfried Bruns and Udo Vetter, Determinantal Rings, LNM1327, Springer1988 · Section2.A, Theorem(2.1), printed p10/PDF18, and proof; Lemma(2.3), printed p11/PDF19.

For a Noetherian ring and proper t-minor ideal of an m by n matrix, its height is at most(m−t+1)(n−t+1).

Hypotheses: Noetherian ring; the ideal must be proper. Apply with A=k[x,y,z], m=2,n=3,t=2.

Reading limits: Read the actual full theorem and proof and the complete supporting Lemma(2.3) proof in the author-hosted PDF. No full-book reading or reconstruction of earlier principal-ideal/dimension inputs is claimed.

Sources