K2 and K3 of the circle

Abstract

Let k be Z[ 1 2 ] , Q or R , and set A = k[x,y] (x 2 + y 2 − 1) . We compute K 2( A) and K 3( A). Our method is to construct a map ϕ : K ∗(k[i])→K ∗ + 1(A) and compare this to a localization sequence. We give three applications. We show that ϕ accounts for the primitive elements in K 2( A), and compare our results to computations of Bloch [1] for group schemes. Secondly, we consider the problem of basepoint independence, and indicate the interplay of geometry upon the K-theory of affine schemes obtained by glueing points of Spec( A). Third, we can iterate the construction to compute the K-theory of the torus ring A ⊗ k A.