Bott periodicity and algebraic geometry

Question

It is a theorem that every locally free coherent sheaf on $\mathbb{P}^1$ over an algebraically closed field is isomorphic to a unique sum of sheaves $\mathcal{O}(n)$ for various integers $n$. In particular, the K-ring of locally free coherent sheaves (or all coherent sheaves, $\mathbb{P}^1$ being nonsingular) is isomorphic to $\mathbb{Z}[t, t^{-1}]$.

The topological K-ring of vector bundles on $S^2$ is, by Bott periodicity, isomorphic to $\mathbb{Z}[H]/(H-1)^2$, where $H$ is the canonical bundle. But $S^2$ is homeomorphic to $\mathbb{P}^1_{\mathbb{C}}$.

Every locally free sheaf corresponds to a vector bundle on $S^2$. It follows that the map on the K-groups from locally free sheaves to vector bundles is surjective but not injective.

Questions:

1. What goes wrong?

2. Is there a version of Bott periodicity for algebraic varieties (or schemes)? (I.e., relating K-groups of $X$ and $X \times \mathbb{P}^1$.) I understand that there is one for the Picard groups.

Answer 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.