K-theory of projective space and of projective bundles

The diagonal section. On $P{\times }_{Y}P$ use the quotient convention of [F1], with $S=\ker (q^{\ast }\mathcal{E}\to \mathcal{O}(1))$ of rank $r$. The composite $S_2\hookrightarrow q^{\ast }\mathcal{E}\twoheadrightarrow \mathcal{O}(1{)}_1$ is a section of $S_2^{\vee }\otimes \mathcal{O}(1{)}_1$; its zero scheme is the diagonal, because the first quotient then factors through the second quotient and the resulting surjection of invertible sheaves is an isomorphism. On a common standard chart where the same coordinate of both quotients is nonzero, the section has regular equations $x_i-y_i$, $1\le i\le r$. These are a regular sequence over every base ring, successively eliminating one coordinate. Such charts cover the diagonal, and off the diagonal one component of the section is a unit so the Koszul complex is contractible. The Koszul complex therefore resolves ${\mathcal{O}}_{\mathrm{\Delta }}$ globally, with terms $\mathcal{O}(-j{)}_1\otimes {\bigwedge }^j{S}_2$.

Pushing the diagonal forward. Tensor the diagonal Koszul resolution with ${\mathrm{pr}}_1^{\ast }\mathcal{F}$ for a coherent sheaf $\mathcal{F}$ on $P$. It remains exact: near the diagonal the equations are coordinate differences in the second factor, a regular sequence on the polynomial extension of every module from the first factor; away from the diagonal it is contractible. Its diagonal term is ${\mathrm{\Delta }}_{\ast }\mathcal{F}$. Apply the alternating coherent pushforward along the proper projection ${\mathrm{pr}}_2$. On an affine open of $Y$ trivializing $\mathcal{E}$, the standard projective Čech complex is bounded of length $r$ and computes all higher direct images; tensoring that complex by a flat algebra commutes with its cohomology. This proves flat base change for the flat map $q$ used here. The projection formula for a locally free factor follows on trivializing opens by finite-direct-sum compatibility of cohomology, as in [F4]. Hence $[\mathcal{F}]={\sum }_{j=0}^r(-1{)}^j{q}^{\ast }{q}_{!}(\mathcal{F}(-j))\otimes [{\bigwedge }^{j}S]$. The higher direct images entering $q_{!}$ are coherent for the proper morphism $q$ and vanish above $r$ by this Čech calculation; the identity is valid on a locally Noetherian base without a ring structure on coherent $K_0(Y)$.

Generation by twists. For $0\to S\to q^{\ast }\mathcal{E}\to \mathcal{O}(1)\to 0$, the exterior-power filtration gives $[{\bigwedge }^j{q}^{\ast }\mathcal{E}]=[{\bigwedge }^{j}S]+[{\bigwedge }^{j-1}S][\mathcal{O}(1)]$. Inducting on $j$ yields $[{\bigwedge }^{j}S]={\sum }_{a=0}^j(-1{)}^a[q^{\ast }{\bigwedge }^{j-a}\mathcal{E}][\mathcal{O}(a)]$. Substitute this into step 2.1 and absorb the pulled-back exterior powers into the coefficient in $K_0(Y)$ via its vector-bundle module action. Every coherent class on $P$ is therefore a sum of $q^{\ast }{\beta }_a\otimes [\mathcal{O}(a)]$, $0\le a\le r$, proving assertion 3.

The trivial bundle and the exterior product. Take $\mathcal{E}={\mathcal{O}}_Y^{n+1}$, so that $P={\mathbb{P}}_k^n{\times }_{k}Y$, the map $q$ is the second projection and $q^{\ast }{K}_0(Y)$ is the image of the exterior product with $[{\mathcal{O}}_{{\mathbb{P}}^n}]$. By step 3.1 the group $K_0({\mathbb{P}}^n{\times }_{k}Y)$ is generated by the classes $q^{\ast }\alpha \otimes [\mathcal{O}(a)]$ with $\alpha \in K_0(Y)$ and $0\le a\le n$, and each such class is the exterior product of a class on $Y$ with $[\mathcal{O}(a)]$ on ${\mathbb{P}}^n$. Hence the exterior-product map $K_0(Y)\otimes K_0({\mathbb{P}}^n)\to K_0({\mathbb{P}}^n{\times }_{k}Y)$ is surjective, which is assertion 2.

The case $Y=\mathrm{Spec}k$ and independence. Let $Y=\mathrm{Spec}k$ and $\mathcal{E}=k^{n+1}$, so $P={\mathbb{P}}_k^n$. Generation by $[\mathcal{O}(0)],\dots ,[\mathcal{O}(n)]$ is step 3.1, and $K^0(P)=K_0(P)$ by [F5] because ${\mathbb{P}}_k^n$ is smooth, projective and hence regular and quasi-projective. For independence suppose ${\sum }_{a=0}^n{c}_a[\mathcal{O}(a)]=0$ in $K_0(P)$. Pairing with the Euler characteristic $\chi (P,-\otimes \mathcal{O}(-j))$, for $0\le j\le n$, is additive on $K_0$, and by [F2] it sends $[\mathcal{O}(a)]$ to $\chi (\mathcal{O}(a-j))=\left(\genfrac{}{}{0px}{}{n+a-j}{n}\right)$ when $a\ge j$ and to $0$ when $a<j$, since then $-n\le a-j\le -1$. The resulting matrix is upper triangular with diagonal entries $\chi (\mathcal{O}(0))=1$, hence invertible over $\mathbb{Z}$, so all $c_a=0$; therefore $[\mathcal{O}(0)],\dots ,[\mathcal{O}(n)]$ is a basis and $K_0({\mathbb{P}}_k^n)\cong {\mathbb{Z}}^{n+1}$, which is assertion 1.