Bott periodicity and algebraic geometry

Solution 1:

It is not true that the Grothendieck ring of coherent sheaves on $\mathbb{P}^1$ is isomorphic to $\mathbb{Z}[t, t^{-1}]$. Although $\mathcal{O} \oplus \mathcal{O}(2)$ is not isomorphic to $\mathcal{O}(1) \oplus \mathcal{O}(1)$, they do have the same class in $K^0$.

The definition of the Grothendieck group of coherent sheaves on a scheme $X$ is that it is generated by isomorphism classes of coherent sheaves, modulo the relation that $[A] + [C] = [B]$ whenever there is a short exact sequence $$0 \to A \to B \to C \to 0.$$ In particular, we have the short exact sequence $$0 \to \mathcal{O} \to \mathcal{O}(1)^2 \to \mathcal{O}(2) \to 0,$$ where the maps are given by $(x \ y)$ and $\binom{-y}{x}$.

This makes $K^0$ into $\mathbb{Z}[t, t^{-1}]/(t^2 - 2t +1) \cong \mathbb{Z}[u]/u^2$, just like you wanted.

When working in the categories of smooth or of topological vector bundles, all short exact sequences split, so you can get away with defining $K$-theory with direct sums. You can't do that in the coherent or the algebraic categories.