Steinberg relations
E911353
Steinberg relations are algebraic identities in Milnor K-theory that impose the condition that symbols of pairs of field elements summing to one vanish, playing a central role in defining the structure of these K-groups.
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic relation
ⓘ
defining relation in Milnor K-theory ⓘ |
| appearsIn |
definition of K_2^{M}(F) as (F^× ⊗ F^×)/R where R is generated by {a,1-a}
ⓘ
higher Milnor K-groups via tensor products modulo Steinberg relations ⓘ |
| appliesTo |
K_2^{M}(F)
ⓘ
K_n^{M}(F) ⓘ Milnor K-groups NERFINISHED ⓘ |
| category |
relation in graded ring
ⓘ
universal relation for symbols on F^× ⓘ |
| condition |
symbols of pairs of field elements summing to one vanish
ⓘ
{a,1-a,a_3, … ,a_n}=0 in K_n^{M}(F) for a,1-a in F^× ⓘ {a,1-a}=0 in K_2^{M}(F) for a,1-a in F^× ⓘ |
| context |
algebraic geometry
ⓘ
algebraic number theory ⓘ motivic cohomology ⓘ |
| definedOver | field F ⓘ |
| field |
Milnor K-theory
NERFINISHED
ⓘ
algebraic K-theory NERFINISHED ⓘ |
| generalizationOf | classical relation in K_2 of a field ⓘ |
| implies |
relations among cross-ratios in K_2^{M}(F)
ⓘ
{a,-a}=0 in K_2^{M}(F) for a in F^× with characteristic not 2 ⓘ |
| inspired | terminology for Steinberg groups in algebraic K-theory ⓘ |
| involves |
Milnor symbols
ⓘ
multiplicative group F^× ⓘ symbols {a,b} in K_2^{M}(F) ⓘ symbols {a_1, … ,a_n} in K_n^{M}(F) ⓘ |
| logicalForm |
for all a in F^× with 1-a in F^×, {a,1-a}=0
ⓘ
for all a_1,a_2 in F^× with a_1+a_2=1, {a_1,a_2}=0 ⓘ |
| namedAfter | Robert Steinberg NERFINISHED ⓘ |
| property |
bilinear in each argument of the Milnor symbol
ⓘ
compatible with graded-commutativity of Milnor K-theory ⓘ functorial with respect to field homomorphisms ⓘ |
| relatedConcept |
Bloch group
NERFINISHED
ⓘ
Matsumoto theorem NERFINISHED ⓘ Milnor K-theory NERFINISHED ⓘ Quillen K-theory NERFINISHED ⓘ universal central extension of Chevalley groups ⓘ |
| role |
define the structure of Milnor K-theory
ⓘ
identify trivial symbols in Milnor K-theory ⓘ impose relations among generators of Milnor K-groups ⓘ |
| usedFor |
defining generators-and-relations description of Milnor K-groups
ⓘ
presentation of K_n^{M}(F) as quotient of tensor powers of F^× ⓘ |
| usedIn |
computations of K_2 of global fields
ⓘ
construction of symbols in Galois cohomology ⓘ regulator maps from K-theory to cohomology ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.