$G_1$ of a nodal projective curve.

Question

Let $C$ be the projective nodal curve over the field $k$ i.e. the projective line with two points identified. I'm having trouble calculating the $G$-theory of $C$, especially $G_1$. Writing the localization sequence wrt closed subscheme the singular point we get: $$G_1(k)\rightarrow G_1(C) \rightarrow G_1(k[t,t^{-1}])\rightarrow G_0(k)\rightarrow G_0( C)\rightarrow G_0(k[t,t^{-1}])\rightarrow 0$$ This reduces to: $$k^{\times}\rightarrow G_1(C) \rightarrow \mathbb{Z}\oplus k^{\times}\rightarrow \mathbb{Z}\rightarrow G_0(C)\rightarrow \mathbb{Z}\rightarrow 0$$ The question is whether $G_0(k)\rightarrow G_0(C)$ is injective or not?

Answer 1

The morphism $G_0(C) \to G_0(k)$ induced by the pushforward via the structure morphism $C \to \mathrm{Spec}(k)$ splits the morphism $G_0(k) \to G_0(C)$, hence the latter is injective.