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.
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.