Nisnevich topology
E884926
The Nisnevich topology is a Grothendieck topology on schemes tailored to capture étale-local algebraic information while ensuring strong local lifting properties over points.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Nisnevich topology canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10773143 — 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: Nisnevich topology Context triple: [Grothendieck topology, generalizes, Nisnevich topology]
-
A.
Grothendieck topology
A Grothendieck topology is an abstract framework in category theory that generalizes the notion of open covers in topology to define sheaves on arbitrary categories.
-
B.
Zariski topology
The Zariski topology is a fundamental topology in algebraic geometry, defined on the spectrum of a ring or an algebraic variety, whose closed sets correspond to solution sets of polynomial equations.
-
C.
Grothendieck’s scheme-theoretic framework
Grothendieck’s scheme-theoretic framework is a foundational reformulation of algebraic geometry that generalizes varieties using schemes, enabling powerful tools like sheaf theory, cohomology, and modern number-theoretic applications.
-
D.
étale cohomology
Étale cohomology is a cohomology theory in algebraic geometry that allows one to apply topological and cohomological methods to schemes, particularly over fields with nontrivial arithmetic such as finite fields.
-
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: Nisnevich topology Target entity description: The Nisnevich topology is a Grothendieck topology on schemes tailored to capture étale-local algebraic information while ensuring strong local lifting properties over points.
-
A.
Grothendieck topology
A Grothendieck topology is an abstract framework in category theory that generalizes the notion of open covers in topology to define sheaves on arbitrary categories.
-
B.
Zariski topology
The Zariski topology is a fundamental topology in algebraic geometry, defined on the spectrum of a ring or an algebraic variety, whose closed sets correspond to solution sets of polynomial equations.
-
C.
Grothendieck’s scheme-theoretic framework
Grothendieck’s scheme-theoretic framework is a foundational reformulation of algebraic geometry that generalizes varieties using schemes, enabling powerful tools like sheaf theory, cohomology, and modern number-theoretic applications.
-
D.
étale cohomology
Étale cohomology is a cohomology theory in algebraic geometry that allows one to apply topological and cohomological methods to schemes, particularly over fields with nontrivial arithmetic such as finite fields.
-
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 |
Grothendieck topology
ⓘ
topology on schemes ⓘ |
| appliesTo |
general schemes
ⓘ
schemes of finite type over a base ⓘ |
| arisesFrom | Nisnevich cd-structure NERFINISHED ⓘ |
| associatedWith |
distinguished Nisnevich squares
ⓘ
excision squares in K-theory ⓘ |
| baseChangeBehavior | stable under base change of schemes ⓘ |
| characterizedBy |
distinguished squares
ⓘ
pointwise lifting property for étale morphisms ⓘ |
| coarserThan | étale topology NERFINISHED ⓘ |
| comparedWith |
Zariski topology
NERFINISHED
ⓘ
étale topology ⓘ |
| compatibleWith | étale-local algebraic information ⓘ |
| coveringFamilyCondition | for every point of the base there exists a point in some cover with isomorphic residue field and mapping to it ⓘ |
| coversGivenBy | families of étale morphisms satisfying residue field isomorphism conditions ⓘ |
| definedOn |
category of schemes
ⓘ
category of schemes over a base scheme ⓘ |
| ensures |
existence of sections after refinement around points
ⓘ
strong local lifting properties over points ⓘ |
| generalizes | Zariski open covers via étale refinements ⓘ |
| hasProperty |
enough points
ⓘ
finer than Zariski but not as fine as étale ⓘ subcanonical ⓘ |
| introducedBy | Yevsey Nisnevich NERFINISHED ⓘ |
| morphismCondition | covers consist of jointly surjective families of étale morphisms with residue field lifting ⓘ |
| motivation | to capture étale-local behavior while improving pointwise lifting properties ⓘ |
| refines | Zariski topology NERFINISHED ⓘ |
| relatedConcept | cd-structure ⓘ |
| supports |
descent for algebraic K-theory
ⓘ
excision in algebraic K-theory ⓘ |
| typicalCover | étale morphism admitting sections over all residue fields of the base ⓘ |
| usedFor |
comparison of algebraic and topological K-theory in certain settings
ⓘ
construction of homotopy invariant sheaves ⓘ localization arguments in motivic homotopy ⓘ |
| usedIn |
A¹-homotopy theory
NERFINISHED
ⓘ
Morel–Voevodsky A¹-homotopy theory NERFINISHED ⓘ Voevodsky’s construction of triangulated categories of motives ⓘ algebraic K-theory ⓘ algebraic geometry ⓘ descent theory ⓘ motivic homotopy theory ⓘ |
| usedToDefine |
Nisnevich cohomology
NERFINISHED
ⓘ
Nisnevich descent NERFINISHED ⓘ Nisnevich sheaves NERFINISHED ⓘ Nisnevich-local model structures on motivic spectra ⓘ Nisnevich-local model structures on simplicial presheaves ⓘ |
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: Nisnevich topology Description of subject: The Nisnevich topology is a Grothendieck topology on schemes tailored to capture étale-local algebraic information while ensuring strong local lifting properties over points.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.