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.