Serre spectral sequence
E256258
The Serre spectral sequence is a fundamental tool in algebraic topology that relates the homology or cohomology of a fibration to that of its base and fiber, enabling complex computations in a systematic way.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Leray–Serre spectral sequence | 2 |
| Serre spectral sequence canonical | 2 |
| homology Serre spectral sequence | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2306391 — 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: Serre spectral sequence Context triple: [Jean-Pierre Serre, notableWork, Serre spectral sequence]
-
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.
Serre duality
Serre duality is a fundamental theorem in algebraic geometry that generalizes classical duality for Riemann surfaces to higher-dimensional projective varieties, relating cohomology groups of coherent sheaves via a dualizing sheaf.
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C.
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.
-
D.
Chern classes
Chern classes are fundamental topological invariants in differential and algebraic geometry that classify complex vector bundles and capture their curvature and twisting properties.
-
E.
Poincaré duality
Poincaré duality is a fundamental theorem in algebraic topology that relates the homology and cohomology groups of an oriented closed manifold in complementary dimensions.
- 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: Serre spectral sequence Target entity description: The Serre spectral sequence is a fundamental tool in algebraic topology that relates the homology or cohomology of a fibration to that of its base and fiber, enabling complex computations in a systematic way.
-
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.
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.
-
C.
Serre duality
Serre duality is a fundamental theorem in algebraic geometry that generalizes classical duality for Riemann surfaces to higher-dimensional projective varieties, relating cohomology groups of coherent sheaves via a dualizing sheaf.
-
D.
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.
-
E.
Chern classes
Chern classes are fundamental topological invariants in differential and algebraic geometry that classify complex vector bundles and capture their curvature and twisting properties.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
spectral sequence ⓘ tool in algebraic topology ⓘ |
| alternativeName |
Leray–Serre spectral sequence
ⓘ
linked to:
Serre spectral sequence
|
| appearsIn | standard textbooks on algebraic topology ⓘ |
| appliesTo |
Serre fibration
ⓘ
fibration of topological spaces ⓘ |
| convergenceType | converges to a filtration of the (co)homology of the total space ⓘ |
| convergesTo | (co)homology of the total space of the fibration ⓘ |
| E2Term |
E2^{p,q} ≅ H^p(B; H^q(F)) in cohomology version
ⓘ
E2_{p,q} ≅ H_p(B; H_q(F)) in homology version ⓘ |
| field |
algebraic topology
ⓘ
homological algebra ⓘ |
| generalizes | Leray spectral sequence in topological setting ⓘ |
| hasAssumption |
base space is path-connected in standard formulations
ⓘ
fiber is path-connected in standard formulations ⓘ |
| hasDifferentials |
dr maps of bidegree (r,1−r) in cohomology version
ⓘ
dr maps of bidegree (−r,r−1) in homology version ⓘ |
| hasStructure | bigraded groups with differentials ⓘ |
| hasVersion |
cohomology Serre spectral sequence
ⓘ
homology Serre spectral sequence ⓘ
linked to:
Serre spectral sequence
|
| introducedBy | Jean-Pierre Serre ⓘ |
| introducedIn | 20th century ⓘ |
| isToolFor | inductive calculations on skeleta of CW-complexes ⓘ |
| namedAfter | Jean-Pierre Serre ⓘ |
| pageE2 | E2 page is expressed in terms of (co)homology of base and fiber ⓘ |
| relatedConcept |
Atiyah–Hirzebruch spectral sequence
ⓘ
Leray–Serre spectral sequence ⓘ
linked to:
Serre spectral sequence
|
| relates |
cohomology of a fibration
ⓘ
cohomology of the base space ⓘ cohomology of the fiber ⓘ homology of a fibration ⓘ homology of the base space ⓘ homology of the fiber ⓘ |
| requires |
local coefficient systems in general form
ⓘ
spectral sequence formalism ⓘ |
| standardReference | Jean-Pierre Serre’s original papers on homotopy groups and fibrations ⓘ |
| usedFor |
computing cohomology groups
ⓘ
computing homology groups ⓘ computing homotopy-invariant information of fibrations ⓘ |
| usedIn |
algebraic K-theory
ⓘ
computation of cohomology of principal bundles ⓘ computation of homology of fiber bundles ⓘ computation of homology of loop spaces ⓘ rational homotopy theory ⓘ stable homotopy theory ⓘ |
| usedToProve | Serre’s finiteness theorem for homotopy groups of spheres (via related methods) ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Serre spectral sequence Description of subject: The Serre spectral sequence is a fundamental tool in algebraic topology that relates the homology or cohomology of a fibration to that of its base and fiber, enabling complex computations in a systematic way.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Serre spectral sequence
linked to: Serre spectral sequence
linked to: Serre spectral sequence