universal coefficient theorem
E911366
The universal coefficient theorem is a fundamental result in algebraic topology that relates the homology or cohomology groups of a space with coefficients in an arbitrary abelian group to those with integer coefficients via Ext and Tor functors.
All labels observed (2)
| Label | Occurrences |
|---|---|
| universal coefficient theorem canonical | 5 |
| universal coefficient theorem for cohomology | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11219623 — 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: universal coefficient theorem Context triple: [Characteristic Classes, hasSubject, universal coefficient theorem]
-
A.
Künneth formula
The Künneth formula is a fundamental result in algebraic topology and homological algebra that expresses the (co)homology of a product space or object in terms of the (co)homology of its factors.
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B.
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.
-
C.
Tate cohomology
Tate cohomology is a refinement of group cohomology that extends cohomology and homology to negative degrees, playing a key role in algebraic number theory and Galois cohomology.
-
D.
Alexander–Spanier cohomology
Alexander–Spanier cohomology is a cohomology theory in algebraic topology defined using cochains on all finite subsets of a space, notable for its generality and close relationship to Čech and singular cohomology.
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E.
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.
- 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: universal coefficient theorem Target entity description: The universal coefficient theorem is a fundamental result in algebraic topology that relates the homology or cohomology groups of a space with coefficients in an arbitrary abelian group to those with integer coefficients via Ext and Tor functors.
-
A.
Künneth formula
The Künneth formula is a fundamental result in algebraic topology and homological algebra that expresses the (co)homology of a product space or object in terms of the (co)homology of its factors.
-
B.
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.
-
C.
Tate cohomology
Tate cohomology is a refinement of group cohomology that extends cohomology and homology to negative degrees, playing a key role in algebraic number theory and Galois cohomology.
-
D.
Alexander–Spanier cohomology
Alexander–Spanier cohomology is a cohomology theory in algebraic topology defined using cochains on all finite subsets of a space, notable for its generality and close relationship to Čech and singular cohomology.
-
E.
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.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf | theorem in algebraic topology ⓘ |
| appliesTo |
CW complexes
ⓘ
chain complexes of abelian groups ⓘ singular cohomology ⓘ singular homology ⓘ topological spaces with reasonable finiteness conditions ⓘ |
| dependsOn |
exactness properties of Hom functor
ⓘ
exactness properties of tensor product ⓘ |
| expresses |
cohomology with coefficients as combination of Hom and Ext terms
ⓘ
homology with coefficients as combination of tensor and Tor terms ⓘ |
| field | algebraic topology ⓘ |
| generalizedTo | universal coefficient spectral sequence ⓘ |
| hasConsequence |
computation of cohomology with finite coefficients
ⓘ
computation of homology with finite coefficients ⓘ reduction of coefficient computations to integer homology ⓘ |
| hasProperty |
expresses universal behavior of coefficient change
ⓘ
functorial in the coefficient group ⓘ natural with respect to continuous maps ⓘ |
| hasVersion |
universal coefficient theorem for cohomology
ⓘ
linked to:
universal coefficient theorem
universal coefficient theorem for generalized homology theories ⓘ universal coefficient theorem for homology ⓘ universal coefficient theorem for reduced homology ⓘ |
| involves |
Hom functor
ⓘ
derived functors ⓘ short exact sequences ⓘ tensor product of abelian groups ⓘ |
| isRelatedTo |
Ext groups
ⓘ
Hurewicz theorem ⓘ
linked to:
Hurewicz homomorphism
Künneth theorem ⓘ
linked to:
Künneth formula
Tor groups ⓘ free resolutions ⓘ homological algebra ⓘ projective resolutions ⓘ |
| relates |
cohomology groups with coefficients in an abelian group
ⓘ
cohomology groups with integer coefficients ⓘ homology groups with coefficients in an abelian group ⓘ homology groups with integer coefficients ⓘ |
| typicalDomain |
abelian groups
ⓘ
modules over a principal ideal domain ⓘ |
| usedIn |
K-theory
ⓘ
classification of topological spaces up to homology ⓘ cohomology operations ⓘ computations in algebraic topology ⓘ spectral sequence calculations ⓘ stable homotopy theory ⓘ |
| usesFunctor |
Ext functor
ⓘ
Tor functor ⓘ |
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: universal coefficient theorem Description of subject: The universal coefficient theorem is a fundamental result in algebraic topology that relates the homology or cohomology groups of a space with coefficients in an arbitrary abelian group to those with integer coefficients via Ext and Tor functors.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.