Projective bundle theorem

Remarks

For $n = 0$ the theorem says that $K_0(\mathbb{P}(\mathcal{E}))$ is a free $K_0(X)$-module with basis $1, [\mathcal{O}(-1)], \ldots, [\mathcal{O}(-r)]$. Quillen proved the statement for all $n$. His proof uses Castelnuovo–Mumford regularity: it gives functorial finite resolutions of vector bundles on $\mathbb{P}(\mathcal{E})$ whose terms are sums of sheaves $\mathcal{O}(-i) \otimes \pi^* \mathcal{F}_i$ with $0 \le i \le r$.

Example: for $X = \operatorname{Spec} k$ with $k$ a field, $K_0(\mathbb{P}^r_k) \cong \Z^{r+1}$. As a ring it is $\Z[t]/(1 - t)^{r+1}$ with $t = [\mathcal{O}(-1)]$; the relation comes from the Koszul resolution of the empty intersection of $r + 1$ hyperplanes.

Going deeper

For a student

The theorem is one of the basic computations of K-theory of schemes, with the localization sequence and homotopy invariance. It parallels the projective bundle formula for Chow groups and for cohomology, where the powers of the first Chern class of $\mathcal{O}(1)$ play the role of the classes $[\mathcal{O}(-i)]$. It is a key step in the proof of the Grothendieck–Riemann–Roch theorem, where a projective morphism is factored through a projective bundle.