Grothendieck spectral sequence

E254133

mathematical concept spectral sequence tool in homological algebra

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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All labels observed (2)

Label	Occurrences
Grothendieck spectral sequence canonical	1
Leray spectral sequence	1

Statements (47)

Predicate	Object
instanceOf	mathematical concept ⓘ spectral sequence ⓘ tool in homological algebra ⓘ
appearsIn	Éléments de géométrie algébrique ⓘ surface form: EGA (Éléments de Géométrie Algébrique) Séminaire de Géométrie Algébrique du Bois Marie ⓘ surface form: SGA (Séminaire de Géométrie Algébrique)
appliesTo	composition of left exact functors ⓘ derived functors in abelian categories ⓘ
assumes	existence of enough injectives ⓘ left exactness of functors ⓘ
category	homological spectral sequences ⓘ
context	cohomological algebra ⓘ derived functor formalism ⓘ
convergesTo	R^{p+q} (G∘F) (A) ⓘ
dependsOn	commutation of functors with injective resolutions ⓘ exactness properties of functors ⓘ
domain	abelian categories ⓘ
field	algebraic geometry ⓘ category theory ⓘ homological algebra ⓘ
formalism	classical derived functors ⓘ derived categories ⓘ
generalizes	Grothendieck spectral sequence self-linksurface differs ⓘ surface form: Leray spectral sequence
hasE2Term	R^p G (R^q F (A)) ⓘ
input	two composable left exact functors F and G ⓘ
namedAfter	Alexander Grothendieck ⓘ
output	spectral sequence converging to derived functors of composite functor G∘F ⓘ
property	compatible with long exact sequences in cohomology ⓘ functorial in the object A ⓘ
purpose	efficient computation of cohomology ⓘ
relatedTo	Cartan–Eilenberg spectral sequence ⓘ Leray spectral sequence ⓘ hypercohomology spectral sequence ⓘ
relates	derived functors of a composite functor ⓘ derived functors of component functors ⓘ
requires	composition of derived functors ⓘ injective resolutions ⓘ
toolFor	breaking complex cohomology computations into simpler stages ⓘ
type	first quadrant spectral sequence in many applications ⓘ
typicalNotation	E_2^{p,q} = R^p G (R^q F (A)) ⇒ R^{p+q} (G∘F) (A) ⓘ
usedFor	computing Ext functors ⓘ computing derived functors of composite functors ⓘ computing group cohomology ⓘ computing sheaf cohomology ⓘ
usedIn	algebraic number theory ⓘ algebraic topology ⓘ representation theory ⓘ
usedToProve	relations between different cohomology theories ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Alexander Grothendieck → notableConcept → Grothendieck spectral sequence ⓘ

Grothendieck spectral sequence → generalizes → Grothendieck spectral sequence self-linksurface differs ⓘ

this entity surface form: Leray spectral sequence