What is wrong in my computation of K groups?

Question

I assume the sequence $0\to C_0(\mathbb{R})\to C(S^1)\to\mathbb{C}\to 0$ is an exact sequence of C*-algebras. Then I suppose we can use the Bott Periodicity, note $K_0(C_0(\mathbb{R}))=0, K_0(C(S^1))=\mathbb{Z}, K_0(\mathbb{C})=\mathbb{Z}, K_1(C_0(\mathbb{R}))=\mathbb{Z},K_1(C(S^1))=\mathbb{Z}, K_1(\mathbb{C})=0$, thus we get the sequence $0\to\mathbb{Z}\to\mathbb{Z}\to\mathbb{Z}\to\mathbb{Z}\to 0$, but I can hardly see why this sequence can be exact?

Answer 1

You did nothing wrong. For instance, the sequence $$0\to\mathbb{Z}\stackrel{1}{\to}\mathbb{Z}\stackrel{0}{\to}\mathbb{Z}\stackrel{1}{\to}\mathbb{Z}\to 0$$ is exact (and, in fact, if you choose the right isomorphisms of your groups with $\mathbb{Z}$, it is your sequence).