What is the Grothendieck (K_0) group of the ring of Laurent polynomials?

Question

I am curious to know what is the Grothendieck group of the ring of Laurent polynomials over a field. I am a beginner in the study of Algebraic K-theory. I have learned that the Grothendieck group of any PID and local ring is the group of integers. Do the same thing happens for ring of Laurent polynomials ? Please anybody help me by giving a description. Thanks in advance.

Answer 1

The fundamental theorem for $K_*$ gives $K_*(R[t, 1/t])$ in terms of $K_*(R[t])$ and $K_*(R)$. The case of $R = k$ a field is simpler and gives $K_0(k[t, t^{-1}]) = K_0(k) = \mathbb{Z}$. (The latter isomorphism, at least, is easy to explain: it's just saying that every f.g. projective module over $k$ is free and classified by its dimension as a vector space, i.e., its rank over $k$.)