Splitting in the K-theory localization sequence of number fields

Luca Caputo (1)

1 School of Mathematical Sciences [Dublin]

Let p be a rational prime and let F be a number field. Then, for each i ≥ 1, Quillen's K-theory group K_{2i}(F ) is a torsion abelian group, containing the finite subgroup K_{2i}(O_F), where O_F is the ring of integers of F . If p is odd or F is nonexceptional or i is even, we give necessary and sufficient conditions for the p-primary component of K_{2i}(O_F) ⊂ K_{2i}(F ) to split. Our conditions involve coinvariants of twisted p-parts of the p-class groups of certain subfields of the fields F(μ_{p^n}) for n ∈ N. We also compare our conditions with the weaker condition WK_{2i}(F) = 0 and give some examples.

Domaines: Théorie des nombres [math.NT]

Luca Caputo. Splitting in the K-theory localization sequence of number fields. Journal of Pure and Applied Algebra, 2010, 215 (4), pp.485-495. ⟨10.1016/j.jpaa.2010.06.001⟩. ⟨hal-00572823⟩