A new exact sequence for 𝐾₂ and some consequences for rings of integers
R. Keith Dennis, Michael R. Stein · Bulletin of the American Mathematical Society · 1972
Suppose R is a Dedekind domain with field of fractions F and at most countably many maximal ideals P. Using methods from the theory of algebraic groups, Bass and Tate [B-T] have proved the exactness of the sequencewhere t is induced by the tame symbols on R.They have also asked whether this sequence remains exact with "0 -•" inserted on the left when R is a ring of algebraic integers.In this note we announce an affirmative response when R is a discrete valuation ring, and a proof that the resulting sequence is split exact under certain additional hypotheses on R. In addition, we derive consequences of these results for a ring, O, of integers in a number field.Among these are (1) a complete determination of the groups K 2 (C/a) for any ideal a of O; and(2) examples of rings of integers O for which K 2 (0) is not generated by symbols and K 2 (2,O) -• K 2 (3,0) is not surjective.Detailed proofs will appear elsewhere.