Witt vectors
E846110
algebraic construction
functor
ring-valued functor
tool in arithmetic geometry
tool in number theory
Witt vectors are algebraic constructions that encode information about rings in characteristic p by packaging sequences of elements into a new ring with specially defined addition and multiplication, widely used in number theory and arithmetic geometry.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic construction
ⓘ
functor ⓘ ring-valued functor ⓘ tool in arithmetic geometry ⓘ tool in number theory ⓘ |
| alsoKnownAs |
Witt ring construction
ⓘ
p-typical Witt vectors NERFINISHED ⓘ |
| appliedTo |
complete discrete valuation rings
ⓘ
finite fields ⓘ perfect fields of characteristic p ⓘ |
| characterizedBy |
Witt polynomials
NERFINISHED
ⓘ
ghost components ⓘ |
| codomainOfFunctor | commutative rings ⓘ |
| constructedFrom |
infinite sequences of ring elements
ⓘ
p-typical Witt components ⓘ |
| defines | Witt ring of a ring R ⓘ |
| domainOfFunctor | commutative rings ⓘ |
| enables | construction of unramified complete discrete valuation rings with given residue field ⓘ |
| encodesInformationAbout |
mod p reductions of rings
ⓘ
rings of characteristic p ⓘ |
| generalizes |
Teichmüller representatives
NERFINISHED
ⓘ
p-adic integers ⓘ |
| hasOperator |
Frobenius
NERFINISHED
ⓘ
Verschiebung ⓘ |
| hasStructure |
functorial ring operations
ⓘ
ring ⓘ unital ring ⓘ |
| hasVariant |
big Witt vectors
ⓘ
ramified Witt vectors ⓘ truncated Witt vectors ⓘ |
| introducedIn | 1930s ⓘ |
| namedAfter | Ernst Witt NERFINISHED ⓘ |
| purpose |
to lift rings of characteristic p to characteristic 0
ⓘ
to study congruence information in a functorial way ⓘ |
| relatedTo |
Dieudonné theory
NERFINISHED
ⓘ
Frobenius endomorphism NERFINISHED ⓘ Verschiebung operator ⓘ unramified extensions of p-adic fields ⓘ |
| usedFor |
constructing p-adic cohomology theories
ⓘ
lifting Frobenius actions ⓘ studying deformation of schemes in characteristic p ⓘ |
| usedIn |
algebraic K-theory
NERFINISHED
ⓘ
arithmetic geometry ⓘ crystalline cohomology ⓘ deformation theory ⓘ number theory ⓘ p-adic Hodge theory NERFINISHED ⓘ theory of formal groups ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.