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
ⓘ
exceptional collection on projective space ⓘ |
| categoryTheoreticContext |
derived functors
ⓘ
triangulated categories ⓘ |
| computes | graded pieces of a resolution of a sheaf ⓘ |
| constructionUses |
Fourier–Mukai transform
ⓘ
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 ⓘ |
| output | complex resolving the sheaf by exceptional objects ⓘ |
| relatedConcept |
Beilinson resolution
ⓘ
linked to:
Beilinson resolution of the diagonal
Bott formula ⓘ
linked to:
Bott residue formula
Fourier–Mukai kernel ⓘ
linked to:
Fourier–Mukai transform
derived category ⓘ exceptional collection ⓘ |
| relatedToSpace | projective space P^n ⓘ |
| requires |
Borel–Bott–Weil theorem on P^n
ⓘ
linked to:
Borel–Weil–Bott theorem
vanishing theorems on projective space ⓘ |
| typicalAmbientSpace | projective n-space over a field ⓘ |
| typicalBaseField | algebraically closed field ⓘ |
| typicalExceptionalCollection | {O(-n), O(-n+1), …, O} ⓘ |
| 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
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Instruction
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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.