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