Relative projective bundles: K-theory generation by tautological twists

The diagonal identity. If $r=0$, $P=\mathrm{\varnothing }$ by the projective-bundle definition, so every class and pushforward is zero and the assertion is immediate; assume $r\ge 1$ from here onward. The section of [L2] meets the stronger regular-section hypothesis: trivialize $E$ over $\mathrm{Spec}A$ and, near a diagonal point, use the same projective chart on both factors with lines generated by $e_0+{\sum }_{i=1}^{r-1}{u}_i{e}_i$ and $e_0+{\sum }_{i=1}^{r-1}{v}_i{e}_i$. Modulo the second line, the first generator has components $u_i-v_i$ in the quotient basis $e_1,\dots ,e_{r-1}$ over $A[u_1,\dots ,u_{r-1},v_1,\dots ,v_{r-1}]$. Successively eliminating $u_i$ makes each next difference monic in a new variable, hence a nonzerodivisor over any base ring; these $r-1$ components form a regular sequence, and the argument survives every base change on $A$. Off the diagonal some component is a unit locally, because the two residue-field lines are distinct. For $r=1$ the sequence is empty and the relative diagonal is the whole product. Thus [L2] supplies the exact Koszul resolution in all cases $r\ge 1$. For a vector bundle $H$ on $P$, tensor the Koszul resolution of the diagonal of [L2] with ${\mathrm{pr}}_1^{\ast }H$ and push forward along ${\mathrm{pr}}_2$: the diagonal term contributes $H$, because ${\mathrm{pr}}_2$ restricts to the identity on the diagonal, and the $j$-th Koszul term contributes $q^{\ast }(q_{!}(H(-j)))\otimes [{\bigwedge }^j{Q}^{\vee }]$ by the projection formula and flat base change [L3]. Hence $[H]={\sum }_{j=0}^{r-1}(-1{)}^j{q}^{\ast }(q_{!}(H(-j))\left)\right[{\bigwedge }^j{Q}^{\vee }]$ in $K^0(P)$.

Generation. The dual universal sequence $0\to Q^{\vee }\to q^{\ast }{E}^{\vee }\to \mathcal{O}(1)\to 0$ gives the exterior-power identity $[{\bigwedge }^j{q}^{\ast }{E}^{\vee }]=[{\bigwedge }^j{Q}^{\vee }]+[{\bigwedge }^{j-1}{Q}^{\vee }][\mathcal{O}(1)]$, by the two graded pieces of the exterior-power filtration in a local splitting. Solving this recurrence, starting with ${\bigwedge }^0{Q}^{\vee }=\mathcal{O}$, gives $[{\bigwedge }^j{Q}^{\vee }]={\sum }_{a=0}^j(-1{)}^a[q^{\ast }{\bigwedge }^{j-a}{E}^{\vee }][\mathcal{O}(a)]$; substituting into step 1.1 shows that every class $[H]$ of a vector bundle is a finite combination of $q^{\ast }{K}^0(Y)$-classes tensored with $\mathcal{O}(a)$, $0\le a<r$. By [L1] every coherent sheaf is resolved by vector bundles, so the same classes generate $K_0(P)$; the action of $q^{\ast }{K}^0(Y)$ is the module structure and this proves generation.

Pushforward of the twists. For $a\ge 0$, the sheaf $q_{!}\mathcal{O}(a)$ is computed on the standard projective affine cover [L3]: the local projective-space cohomology calculation has only degree-zero cohomology with its homogeneous monomial basis of degree $a$, and changes of trivialization act on this basis by ${\mathrm{Sym}}^a(E^{\vee })$; higher direct images vanish by the same computation and the Čech cover comparison. Hence $q_{!}\mathcal{O}(a)={\mathrm{Sym}}^a(E^{\vee })$ as a vector bundle on $Y$, which is the second assertion; the dual statement for quotient projective bundles follows by dualizing $E$ and using $\mathbb{P}(E^{\vee })$ with the induced universal sequence.