Triple

T16151065
Position Surface form Disambiguated ID Type / Status
Subject equivariant index theorem E391908 entity
Predicate relatedTo P37 FINISHED
Object equivariant Riemann–Roch theorem
The equivariant Riemann–Roch theorem is a generalization of the classical Riemann–Roch theorem that computes characteristic classes and indices in the presence of a group action, relating equivariant K-theory to equivariant cohomology.
E1197992 NE FINISHED

How this triple was built (4 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: equivariant Riemann–Roch theorem | Statement: [equivariant index theorem, relatedTo, equivariant Riemann–Roch theorem]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: equivariant Riemann–Roch theorem
Context triple: [equivariant index theorem, relatedTo, equivariant Riemann–Roch theorem]
  • A. Hirzebruch–Riemann–Roch theorem
    The Hirzebruch–Riemann–Roch theorem is a fundamental result in algebraic geometry and topology that expresses the holomorphic Euler characteristic of a complex manifold in terms of characteristic classes, unifying and extending classical Riemann–Roch type formulas.
  • B. Grothendieck–Riemann–Roch theorem
    The Grothendieck–Riemann–Roch theorem is a fundamental result in algebraic geometry that generalizes the classical Riemann–Roch theorem by relating pushforwards in K-theory to pushforwards in cohomology via characteristic classes.
  • C. Grothendieck–Ogg–Shafarevich formula
    The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
  • D. Atiyah–Bott fixed-point theorem
    The Atiyah–Bott fixed-point theorem is a fundamental result in equivariant cohomology that expresses global invariants, such as indices of elliptic operators, in terms of local data at the fixed points of a group action.
  • E. Riemann–Roch theorem
    The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
  • F. None of above. chosen
  • G. Unsure - the case is ambiguous/there is not enough information to decide.
NEDg Description generation gpt-5.1
Instruction
Generate a one-sentence description of the target entity. 
You are given a context triple in the form (subject, predicate, object), where the object is the target entity. 
# Instructions
Use the triple to infer relevant information about the entity. Describe the entity based on what is most defining, well-known. 
Avoid repeating the information from the triple, unless really essential.
# Response Format
Return only the sentence: "Description: [one-sentence description of the target entity]"
Input
Entity: equivariant Riemann–Roch theorem
Triple: [equivariant index theorem, relatedTo, equivariant Riemann–Roch theorem]
Generated description
The equivariant Riemann–Roch theorem is a generalization of the classical Riemann–Roch theorem that computes characteristic classes and indices in the presence of a group action, relating equivariant K-theory to equivariant cohomology.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: equivariant Riemann–Roch theorem
Target entity description: The equivariant Riemann–Roch theorem is a generalization of the classical Riemann–Roch theorem that computes characteristic classes and indices in the presence of a group action, relating equivariant K-theory to equivariant cohomology.
  • A. Hirzebruch–Riemann–Roch theorem
    The Hirzebruch–Riemann–Roch theorem is a fundamental result in algebraic geometry and topology that expresses the holomorphic Euler characteristic of a complex manifold in terms of characteristic classes, unifying and extending classical Riemann–Roch type formulas.
  • B. Grothendieck–Riemann–Roch theorem
    The Grothendieck–Riemann–Roch theorem is a fundamental result in algebraic geometry that generalizes the classical Riemann–Roch theorem by relating pushforwards in K-theory to pushforwards in cohomology via characteristic classes.
  • C. Grothendieck–Ogg–Shafarevich formula
    The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
  • D. Atiyah–Bott fixed-point theorem
    The Atiyah–Bott fixed-point theorem is a fundamental result in equivariant cohomology that expresses global invariants, such as indices of elliptic operators, in terms of local data at the fixed points of a group action.
  • E. Riemann–Roch theorem
    The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
  • F. None of above. chosen

Provenance (5 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_69d87f1c65e48190aa2b4c472e9bafc4 completed April 10, 2026, 4:39 a.m.
NER Named-entity recognition batch_69e21d981950819087fdacc7879dca97 completed April 17, 2026, 11:46 a.m.
NED1 Entity disambiguation (via context triple) batch_69fff7a9ebf08190aa21cdff051f4ba2 completed May 10, 2026, 3:12 a.m.
NEDg Description generation batch_69fff86a556c819096bc008e1ca76e8c completed May 10, 2026, 3:15 a.m.
NED2 Entity disambiguation (via description) batch_69fff926120081909f1042bf3a16ea10 completed May 10, 2026, 3:19 a.m.
Created at: April 10, 2026, 5:01 a.m.