Beilinson spectral sequence
E876107
The Beilinson spectral sequence is a powerful tool in algebraic geometry that reconstructs coherent sheaves on projective space from their cohomology via a resolution by exceptional collections.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Beilinson spectral sequence canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10617354 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Beilinson spectral sequence Context triple: [Alexander Beilinson, knownFor, Beilinson spectral sequence]
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A.
Grothendieck spectral sequence
The Grothendieck spectral sequence is a fundamental tool in homological algebra that relates the derived functors of a composite functor to the derived functors of its components, enabling efficient computation of cohomology.
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B.
Deligne cohomology
Deligne cohomology is a refined cohomology theory in algebraic geometry that combines singular cohomology and differential forms to capture both topological and arithmetic information about complex algebraic varieties.
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C.
Weil cohomology
Weil cohomology is a type of cohomology theory for algebraic varieties that satisfies specific axioms enabling the proof of the Weil conjectures and the development of modern algebraic geometry.
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D.
Beilinson conjectures
Beilinson conjectures are a set of deep conjectures in arithmetic geometry that relate special values of L-functions to algebraic K-theory and motivic cohomology, generalizing phenomena seen in cases like the Birch and Swinnerton-Dyer conjecture.
-
E.
Atiyah–Hirzebruch spectral sequence
The Atiyah–Hirzebruch spectral sequence is a fundamental computational tool in algebraic topology that relates generalized cohomology theories, such as K-theory, to ordinary cohomology, enabling the step-by-step calculation of these invariants from simpler data.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Beilinson spectral sequence Target entity description: The Beilinson spectral sequence is a powerful tool in algebraic geometry that reconstructs coherent sheaves on projective space from their cohomology via a resolution by exceptional collections.
-
A.
Grothendieck spectral sequence
The Grothendieck spectral sequence is a fundamental tool in homological algebra that relates the derived functors of a composite functor to the derived functors of its components, enabling efficient computation of cohomology.
-
B.
Deligne cohomology
Deligne cohomology is a refined cohomology theory in algebraic geometry that combines singular cohomology and differential forms to capture both topological and arithmetic information about complex algebraic varieties.
-
C.
Weil cohomology
Weil cohomology is a type of cohomology theory for algebraic varieties that satisfies specific axioms enabling the proof of the Weil conjectures and the development of modern algebraic geometry.
-
D.
Beilinson conjectures
Beilinson conjectures are a set of deep conjectures in arithmetic geometry that relate special values of L-functions to algebraic K-theory and motivic cohomology, generalizing phenomena seen in cases like the Birch and Swinnerton-Dyer conjecture.
-
E.
Atiyah–Hirzebruch spectral sequence
The Atiyah–Hirzebruch spectral sequence is a fundamental computational tool in algebraic topology that relates generalized cohomology theories, such as K-theory, to ordinary cohomology, enabling the step-by-step calculation of these invariants from simpler data.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
spectral sequence
ⓘ
tool in algebraic geometry ⓘ |
| appearsIn | Beilinson’s paper on coherent sheaves on P^n ⓘ |
| appliesTo |
coherent sheaves
ⓘ
projective space ⓘ |
| assumes | bounded derived category of coherent sheaves is generated by exceptional collection ⓘ |
| basedOn |
Beilinson resolution of the diagonal
NERFINISHED
ⓘ
exceptional collection on projective space ⓘ |
| categoryTheoreticContext |
derived functors
ⓘ
triangulated categories ⓘ |
| computes | graded pieces of a resolution of a sheaf ⓘ |
| constructionUses |
Fourier–Mukai transform
NERFINISHED
ⓘ
resolution of the diagonal on P^n × P^n ⓘ |
| convergesTo | given coherent sheaf ⓘ |
| domain | derived category of coherent sheaves ⓘ |
| E1PageDescribedBy | cohomology groups tensored with dual exceptional objects ⓘ |
| field | algebraic geometry ⓘ |
| functoriality | functorial in morphisms of coherent sheaves ⓘ |
| generalizationOf | classical resolutions of vector bundles on projective space ⓘ |
| generalizedTo |
other Fano varieties with exceptional collections
ⓘ
toric varieties with full exceptional collections ⓘ weighted projective spaces ⓘ |
| input | graded cohomology groups of a coherent sheaf ⓘ |
| inspiredFurtherWork |
noncommutative projective geometry
ⓘ
semiorthogonal decompositions ⓘ tilting bundles on projective varieties ⓘ |
| namedAfter | Alexander Beilinson NERFINISHED ⓘ |
| output | complex resolving the sheaf by exceptional objects ⓘ |
| relatedConcept |
Beilinson resolution
NERFINISHED
ⓘ
Bott formula NERFINISHED ⓘ Fourier–Mukai kernel NERFINISHED ⓘ derived category ⓘ exceptional collection ⓘ |
| relatedToSpace | projective space P^n GENERATED ⓘ |
| requires |
Borel–Bott–Weil theorem on P^n
NERFINISHED
ⓘ
vanishing theorems on projective space ⓘ |
| typicalAmbientSpace | projective n-space over a field ⓘ |
| typicalBaseField | algebraically closed field GENERATED ⓘ |
| typicalExceptionalCollection | {O(-n), O(-n+1), …, O} GENERATED ⓘ |
| usedBy |
algebraic geometers
ⓘ
homological algebraists ⓘ representation theorists ⓘ |
| usedFor |
computing sheaf cohomology on projective space
ⓘ
describing resolutions by exceptional collections ⓘ reconstruction of coherent sheaves from cohomology ⓘ |
| yearIntroducedApprox | 1978 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Beilinson spectral sequence Description of subject: The Beilinson spectral sequence is a powerful tool in algebraic geometry that reconstructs coherent sheaves on projective space from their cohomology via a resolution by exceptional collections.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.