Triple

T9297133
Position Surface form Disambiguated ID Type / Status
Subject Auslander–Buchsbaum formula E223665 entity
Predicate statement P4223 FINISHED
Object For a Noetherian local ring (R, m) and a finitely generated R-module M of finite projective dimension, pd_R(M) + depth_R(M) = depth(R). E223665 NE FINISHED

How this triple was built (2 steps)

Every LLM step that produced this triple, in pipeline order — named-entity classification, the disambiguation choices (the exact options shown, with the pick highlighted), and the generated description. The batch + timestamp of each is in the Provenance table below.

NER Named-entity recognition gpt-5-mini
Instruction
Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).
Input
Phrase: For a Noetherian local ring (R, m) and a finitely generated R-module M of finite projective dimension, pd_R(M) + depth_R(M) = depth(R). | Statement: [Auslander–Buchsbaum formula, statement, For a Noetherian local ring (R, m) and a finitely generated R-module M of finite projective dimension, pd_R(M) + depth_R(M) = depth(R).]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07 
Target entity: For a Noetherian local ring (R, m) and a finitely generated R-module M of finite projective dimension, pd_R(M) + depth_R(M) = depth(R).
Context triple: [Auslander–Buchsbaum formula, statement, For a Noetherian local ring (R, m) and a finitely generated R-module M of finite projective dimension, pd_R(M) + depth_R(M) = depth(R).]
  • A. Noetherian module
    A Noetherian module is an algebraic structure in which every ascending chain of submodules stabilizes, ensuring that all submodules are finitely generated and enabling powerful finiteness arguments in ring and module theory.
  • B. Auslander–Buchsbaum formula chosen
    The Auslander–Buchsbaum formula is a fundamental result in commutative algebra that relates the projective dimension of a finitely generated module over a Noetherian local ring to the depth of the module and the depth of the ring.
  • C. Noetherian rings
    Noetherian rings are a fundamental class of rings in commutative algebra characterized by the property that every ascending chain of ideals stabilizes, ensuring that all ideals are finitely generated.
  • D. Krull dimension
    Krull dimension is a fundamental invariant in commutative algebra that measures the "size" of a ring by the maximum length of chains of its prime ideals.
  • E. Hilbert’s syzygy theorem
    Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
  • F. None of above.
  • G. Unsure - the case is ambiguous/there is not enough information to decide.

Provenance (3 batches)

The batch behind each pipeline step, in order, with when it ran. Timestamps are batch-level — stages were processed in waves, so the object chain (NER → NED1 → NEDg → NED2) reads in order, but predicate / elicitation batches can sit in a different wave.

Step Stage Batch ID Status When
creating Elicitation batch_69ca8423edb08190bc0c91287a484768 completed March 30, 2026, 2:09 p.m.
NER Named-entity recognition batch_69cd089e3ae88190aa4181cdd85a67b8 completed April 1, 2026, 11:59 a.m.
NED1 Entity disambiguation (via context triple) batch_69d0b251c4148190a94fafdc23a601d6 completed April 4, 2026, 6:40 a.m.
Created at: March 30, 2026, 7:36 p.m.