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