Henselization
E483409
Henselization is a construction in commutative algebra that minimally modifies a local ring to satisfy Hensel’s lemma, making it “Henselian” while preserving much of its original structure.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
construction in commutative algebra
ⓘ
localization-like construction ⓘ universal property construction ⓘ |
| appliesTo | local ring ⓘ |
| categoryTheoreticProperty | left adjoint to the inclusion of Henselian local rings into all local rings in many settings ⓘ |
| comparedTo | completion is typically larger than Henselization ⓘ |
| construction |
can be described as a filtered colimit of étale algebras
ⓘ
often defined via étale neighborhoods ⓘ |
| context |
local algebra
ⓘ
valuation theory ⓘ |
| differsFrom | strict Henselization, which also makes the residue field separably closed ⓘ |
| ensures |
certain finite étale extensions split as products corresponding to residue field extensions
ⓘ
lifting of idempotents from residue field to the ring ⓘ lifting of simple roots of polynomials from residue field to the ring ⓘ uniqueness of factorization of polynomials near simple roots ⓘ uniqueness of lifted roots under Hensel's lemma hypotheses ⓘ |
| extensionProperty |
induces an isomorphism on residue fields
ⓘ
is local and faithfully flat over the original ring in many contexts ⓘ |
| field | commutative algebra ⓘ |
| functoriality | defines a functor from local rings to Henselian local rings ⓘ |
| goal | to make a given local ring Henselian ⓘ |
| introducedIn | theory of Henselian rings ⓘ |
| mapsTo | completion of the local ring via a local homomorphism ⓘ |
| minimality | smallest Henselian local ring containing the original ring with same residue field ⓘ |
| modifies | a local ring minimally to become Henselian ⓘ |
| namedAfter | Kurt Hensel NERFINISHED ⓘ |
| preserves |
completion map injectivity on residue field
ⓘ
dimension of the local ring ⓘ maximal ideal of the local ring ⓘ residue field of the local ring ⓘ |
| produces | Henselian local ring ⓘ |
| property |
is initial among Henselian local rings receiving a local homomorphism from the given ring
ⓘ
often strictly smaller than the completion ⓘ |
| relatedConcept | strict Henselization ⓘ |
| relatedTo | completion of a local ring ⓘ |
| satisfies | Hensel's lemma ⓘ |
| technicalRole | intermediate between a local ring and its completion ⓘ |
| typeOf | local ring extension ⓘ |
| universalProperty | any local homomorphism from the original ring to a Henselian local ring factors uniquely through its Henselization ⓘ |
| usedFor |
constructing Henselian local schemes
ⓘ
local study of schemes near a point ⓘ simplifying lifting problems for polynomial equations ⓘ |
| usedIn |
algebraic geometry
ⓘ
deformation theory ⓘ number theory ⓘ étale cohomology ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.