Cartan–Eilenberg spectral sequence
E884929
The Cartan–Eilenberg spectral sequence is a fundamental tool in homological algebra that computes derived functors (such as Ext and Tor) of composite functors via a double complex construction.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Cartan–Eilenberg spectral sequence canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10773292 — 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: Cartan–Eilenberg spectral sequence Context triple: [Grothendieck spectral sequence, relatedTo, Cartan–Eilenberg 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.
-
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 spectral sequence
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.
-
D.
Beilinson spectral sequence
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.
-
E.
Mayer–Vietoris sequence in de Rham cohomology
The Mayer–Vietoris sequence in de Rham cohomology is a long exact sequence that computes the de Rham cohomology of a manifold by relating it to the cohomology of an open cover and their intersection.
- 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: Cartan–Eilenberg spectral sequence Target entity description: The Cartan–Eilenberg spectral sequence is a fundamental tool in homological algebra that computes derived functors (such as Ext and Tor) of composite functors via a double complex construction.
-
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 spectral sequence
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.
-
D.
Beilinson spectral sequence
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.
-
E.
Mayer–Vietoris sequence in de Rham cohomology
The Mayer–Vietoris sequence in de Rham cohomology is a long exact sequence that computes the de Rham cohomology of a manifold by relating it to the cohomology of an open cover and their intersection.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
spectral sequence ⓘ tool in homological algebra ⓘ |
| appearsIn | Cartan and Eilenberg’s book "Homological Algebra" NERFINISHED ⓘ |
| appliesTo |
abelian categories
ⓘ
chain complexes ⓘ |
| assumes | enough injectives or projectives in the abelian category ⓘ |
| computes |
cohomology of total complex of a double complex
ⓘ
homology of total complex of a double complex ⓘ |
| constructionMethod | double complex ⓘ |
| convergenceType | convergence to the derived functor of the composite functor ⓘ |
| convergesTo | derived functor of the composite functor ⓘ |
| domain |
category theory
ⓘ
homological algebra ⓘ |
| field | homological algebra ⓘ |
| generalizationOf | spectral sequence of a filtered complex ⓘ |
| hasE2Term | derived functors of one functor applied to derived functors of another functor ⓘ |
| hasInput |
composite functor
ⓘ
double complex of objects in an abelian category ⓘ |
| hasPage | E2-page ⓘ |
| hasPrerequisite |
knowledge of chain complexes
ⓘ
knowledge of derived functors ⓘ knowledge of spectral sequences ⓘ |
| isToolFor |
computing cohomology of composite functors
ⓘ
computing homology of composite functors ⓘ |
| mathematicsSubjectClassification |
18G10
ⓘ
18G40 ⓘ |
| namedAfter |
Henri Cartan
NERFINISHED
ⓘ
Samuel Eilenberg NERFINISHED ⓘ |
| relatedTo | Grothendieck spectral sequence for derived functors NERFINISHED ⓘ |
| relatesConcept |
Ext functor
ⓘ
Grothendieck spectral sequence NERFINISHED ⓘ Tor functor ⓘ derived functor ⓘ double complex spectral sequence ⓘ |
| requires | bicomplex or double complex structure ⓘ |
| typicalStatement | there is a spectral sequence with E2-term given by derived functors of one functor applied to derived functors of another ⓘ |
| usedFor |
computing Ext functors
ⓘ
computing Tor functors ⓘ computing derived functors of composite functors ⓘ |
| usedIn |
algebraic geometry
ⓘ
algebraic topology ⓘ group cohomology ⓘ module theory ⓘ |
| usedToProve | relations between Ext and Tor of composite functors ⓘ |
| yearIntroduced | 1950s ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Cartan–Eilenberg spectral sequence Description of subject: The Cartan–Eilenberg spectral sequence is a fundamental tool in homological algebra that computes derived functors (such as Ext and Tor) of composite functors via a double complex construction.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.