Grothendieck spectral sequence
E254133
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Grothendieck spectral sequence canonical | 3 |
| Grothendieck spectral sequence for derived functors | 1 |
| Leray spectral sequence | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2290654 — 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: Grothendieck spectral sequence Context triple: [Alexander Grothendieck, notableConcept, Grothendieck spectral sequence]
-
A.
Alexandrov–Čech cohomology
Alexandrov–Čech cohomology is a topological cohomology theory that computes invariants of spaces using inverse limits over open covers, closely related to and often coinciding with sheaf cohomology.
-
B.
Categories for the Working Mathematician
Categories for the Working Mathematician is a foundational textbook in category theory that systematically develops the subject and its applications for professional mathematicians.
-
C.
Sheaves in Geometry and Logic
Sheaves in Geometry and Logic is a foundational monograph that develops the theory of sheaves and topos theory and explores their deep connections to geometry, logic, and the foundations of mathematics.
-
D.
Atiyah–Singer index theorem
The Atiyah–Singer index theorem is a fundamental result in mathematics that links the analytical properties of elliptic differential operators to topological invariants of manifolds, unifying analysis, topology, and geometry.
-
E.
Weil cohomology
Weil cohomology is a type of cohomology theory for algebraic varieties that satisfies specific axioms enabling the proof of the Weil conjectures and the development of modern algebraic geometry.
- 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: Grothendieck spectral sequence Target entity description: 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.
-
A.
Alexandrov–Čech cohomology
Alexandrov–Čech cohomology is a topological cohomology theory that computes invariants of spaces using inverse limits over open covers, closely related to and often coinciding with sheaf cohomology.
-
B.
Categories for the Working Mathematician
Categories for the Working Mathematician is a foundational textbook in category theory that systematically develops the subject and its applications for professional mathematicians.
-
C.
Sheaves in Geometry and Logic
Sheaves in Geometry and Logic is a foundational monograph that develops the theory of sheaves and topos theory and explores their deep connections to geometry, logic, and the foundations of mathematics.
-
D.
Atiyah–Singer index theorem
The Atiyah–Singer index theorem is a fundamental result in mathematics that links the analytical properties of elliptic differential operators to topological invariants of manifolds, unifying analysis, topology, and geometry.
-
E.
Weil cohomology
Weil cohomology is a type of cohomology theory for algebraic varieties that satisfies specific axioms enabling the proof of the Weil conjectures and the development of modern algebraic geometry.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
spectral sequence ⓘ tool in homological algebra ⓘ |
| appearsIn |
EGA (Éléments de Géométrie Algébrique)
ⓘ
linked to:
Éléments de géométrie algébrique
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 |
Leray spectral sequence
ⓘ
linked to:
Grothendieck 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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Grothendieck spectral sequence Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Grothendieck spectral sequence
Cartan–Eilenberg spectral sequence
→
relatedTo
→
Grothendieck spectral sequence for derived functors
ⓘ
linked to: Grothendieck spectral sequence