Projective Bundle Theorem

The projective bundle theorem computes the K-theory of a projective bundle $\mathbb{P}(\mathcal{E})$ in terms of the K-theory of the base. It is one of the key structural results, generalizing the computation of $K_*(\mathbb{P}^n)$ and providing the foundation for Chern character calculations and Grothendieck's approach to Riemann--Roch.

Statement

Theorem3.6Projective Bundle Theorem

Let $X$ be a Noetherian scheme and $\mathcal{E}$ a locally free sheaf of rank $r$ on $X$ . Let $\pi: \mathbb{P}(\mathcal{E}) \to X$ be the associated projective bundle with tautological line bundle $\mathcal{O}(1)$ . Then the map

$\bigoplus_{i=0}^{r-1} K_n(X) \xrightarrow{\;\sim\;} K_n(\mathbb{P}(\mathcal{E}))$

defined by $(a_0, a_1, \ldots, a_{r-1}) \mapsto \sum_{i=0}^{r-1} \pi^*(a_i) \cdot [\mathcal{O}(i)]$ is an isomorphism for all $n \geq 0$ .

Equivalently, $K_n(\mathbb{P}(\mathcal{E})) \cong K_n(X)^r$ as an abelian group for each $n$ .

Special case: projective space

ExampleK-theory of projective space

Taking $X = \operatorname{Spec} k$ and $\mathcal{E} = \mathcal{O}^{n+1}$ gives $\mathbb{P}(\mathcal{E}) = \mathbb{P}^n_k$ :

$K_m(\mathbb{P}^n_k) \cong K_m(k)^{n+1}$

with basis $[\mathcal{O}], [\mathcal{O}(1)], \ldots, [\mathcal{O}(n)]$ over $K_m(k)$ .

For $m = 0$ and $k$ a field: $K_0(\mathbb{P}^n) \cong \mathbb{Z}^{n+1}$ with generators $1, h, h^2, \ldots, h^n$ where $h = [\mathcal{O}(1)] - [\mathcal{O}]$ . The ring structure on $K_0$ is:

$K_0(\mathbb{P}^n) \cong \mathbb{Z}[h] / (h^{n+1})$

since $[\mathcal{O}(1)]$ acts by the hyperplane class and $([\mathcal{O}(1)] - 1)^{n+1} = 0$ .

ExampleK-theory of Grassmannians

For the Grassmannian $\operatorname{Gr}(r, n) = \operatorname{Gr}(r, k^n)$ parametrizing $r$ -dimensional subspaces of $k^n$ , iterated application of the projective bundle theorem gives:

$K_0(\operatorname{Gr}(r, n)) \cong \mathbb{Z}^{\binom{n}{r}}$

as a free abelian group. The ring structure is described by the representation ring of $GL_r$ : there is an isomorphism $K_0(\operatorname{Gr}(r, n)) \cong R(GL_r) / I$ where $I$ encodes the relations from the ambient $k^n$ .

The Schur functors $\Sigma^{\lambda} \mathcal{S}$ (applied to the tautological bundle $\mathcal{S}$ ) provide a basis indexed by Young diagrams $\lambda$ fitting in an $r \times (n-r)$ box.

Proof sketch

Proof

The proof uses Quillen's localization sequence and induction on the rank $r$ .

Step 1: Base case $r = 1$ . When $\mathcal{E}$ has rank 1, $\mathbb{P}(\mathcal{E}) = X$ and the theorem is trivial.

Step 2: The Koszul filtration. On $\mathbb{P}(\mathcal{E})$ , there is a tautological exact sequence

$0 \to \mathcal{O}(-1) \to \pi^*\mathcal{E} \to \mathcal{Q} \to 0$

where $\mathcal{Q}$ is the universal quotient bundle of rank $r - 1$ . Define the functor $\Phi: K_n(X)^r \to K_n(\mathbb{P}(\mathcal{E}))$ by

$\Phi(a_0, \ldots, a_{r-1}) = \sum_{i=0}^{r-1} \pi^*(a_i) \cdot [\mathcal{O}(i)].$

Step 3: Resolution of the diagonal. Consider $\mathbb{P}(\mathcal{E}) \times_X \mathbb{P}(\mathcal{E})$ with the two projections $p_1, p_2$ and the relative diagonal $\Delta \hookrightarrow \mathbb{P}(\mathcal{E}) \times_X \mathbb{P}(\mathcal{E})$ . The Koszul complex provides a resolution of the structure sheaf of the diagonal:

$0 \to p_1^*\mathcal{O}(-(r-1)) \otimes p_2^*\Omega^{r-1}(r-1) \to \cdots \to p_1^*\mathcal{O}(-1) \otimes p_2^*\Omega^1(1) \to \mathcal{O} \to \mathcal{O}_\Delta \to 0$

This resolution shows that the identity functor on $D^b(\operatorname{Coh}(\mathbb{P}(\mathcal{E})))$ is generated by $\pi^*(-) \otimes \mathcal{O}(i)$ for $i = 0, \ldots, r-1$ .

Step 4: Surjectivity. Every coherent sheaf on $\mathbb{P}(\mathcal{E})$ has a finite resolution by sheaves of the form $\pi^*\mathcal{F} \otimes \mathcal{O}(i)$ . This follows from Serre's theorem (for $i \gg 0$ , $\mathcal{F}(i)$ is generated by global sections that come from the base) combined with the Koszul resolution. This shows $\Phi$ is surjective.

Step 5: Injectivity. Define $\Psi: K_n(\mathbb{P}(\mathcal{E})) \to K_n(X)^r$ using the higher direct images:

$\Psi(\alpha) = (R^0\pi_*(\alpha \otimes \mathcal{O}(-i)))_{i=0}^{r-1}.$

The projection formula $R^j\pi_*(\pi^*\mathcal{F} \otimes \mathcal{O}(k)) = \mathcal{F} \otimes R^j\pi_*\mathcal{O}(k)$ and the computation $R^j\pi_*\mathcal{O}(k) = \begin{cases} S^k\mathcal{E} & j = 0, k \geq 0 \\ 0 & j = 0, -r < k < 0 \end{cases}$ show $\Psi \circ \Phi = \operatorname{id}$ (upper triangular with 1's on the diagonal), proving injectivity. $\square$

■

Consequences

RemarkGrothendieck's gamma filtration and Chern classes

The projective bundle theorem enables the construction of Chern classes in K-theory. For a vector bundle $\mathcal{E}$ of rank $r$ on $X$ , define the Chern classes $c_i(\mathcal{E}) \in K_0(X)$ via the relation:

$\sum_{i=0}^{r} (-1)^i c_i(\mathcal{E}) \cdot [\mathcal{O}(1)]^{r-i} = 0 \quad \text{in } K_0(\mathbb{P}(\mathcal{E}))$

where we view $K_0(\mathbb{P}(\mathcal{E}))$ as a $K_0(X)$ -module. This is the splitting principle: the Chern classes are determined by the behavior of $\mathcal{E}$ on its own projectivization.

The Chern character $\operatorname{ch}: K_0(X) \to CH^*(X) \otimes \mathbb{Q}$ constructed from these operations is a ring homomorphism that becomes an isomorphism after tensoring with $\mathbb{Q}$ :

$K_0(X) \otimes \mathbb{Q} \xrightarrow[\sim]{\operatorname{ch}} CH^*(X) \otimes \mathbb{Q}$

for smooth varieties $X$ over a field.