Riemann-Roch for projective-space projections

K-theory pushforward of the generators. By [L1] it suffices to verify the identity on the generators $p_Y^{\ast }\beta \otimes p_{\mathbb{P}}^{\ast }[\mathcal{O}(m)]$ for $\beta \in K_0(Y)$ and $0\le m\le n$. The K-theory projection formula gives $p_{!}(p^{\ast }\beta \otimes [\mathcal{O}(m)])=\beta \otimes p_{!}[\mathcal{O}(m)]$, and the standard projective Čech calculation on each affine open $U\subseteq Y$ gives $p_{\ast }\mathcal{O}(m){\mathrm{\mid }}_U={\mathcal{O}}_U[x_0,\dots ,x_n{]}_m$ and $R^q{p}_{\ast }\mathcal{O}(m){\mathrm{\mid }}_U=0$ for $q>0$. The same globally fixed homogeneous monomials of degree $m$ give a basis on every $U$, and these bases agree on overlaps because the projective-space factor is constant. Thus $p_{\ast }\mathcal{O}(m)$ is the trivial bundle of rank $\left(\genfrac{}{}{0px}{}{m+n}{n}\right)$, not merely a bundle with that fibre rank; hence $p_{!}[\mathcal{O}(m)]=\left(\genfrac{}{}{0px}{}{m+n}{n}\right)[{\mathcal{O}}_Y]$.

The fibre computation. Let $\tau =\mathrm{ch}\cdot \mathrm{td}$; for the generator $\alpha =p_Y^{\ast }\beta \otimes p_{\mathbb{P}}^{\ast }[\mathcal{O}(m)]$ the left-hand side is $\tau (p_{!}\alpha )=\left(\genfrac{}{}{0px}{}{m+n}{n}\right)\mathrm{ch}(\beta )\mathrm{td}(T_Y)$ by step 1.1 and additivity/multiplicativity of $\tau$ on the base [L2]. The right-hand side is $p_{\ast }(\mathrm{ch}(\beta )\mathrm{ch}(\mathcal{O}(m))\mathrm{td}(T_{{\mathbb{P}}^n{\times }_{k}Y}))$; the tangent bundle of the product is the direct sum of the pullbacks of the tangent bundles, so its Todd class is $\mathrm{td}(T_{{\mathbb{P}}^n})\mathrm{td}(T_Y)$ by [L2], and the projection formula for $p_{\ast }$ [L3] moves $\mathrm{ch}(\beta )\mathrm{td}(T_Y)$ out, leaving the fibre integral $p_{\ast }(\mathrm{ch}(\mathcal{O}(m))\mathrm{td}(T_{{\mathbb{P}}^n}))$. The Euler sequence $0\to \mathcal{O}\to \mathcal{O}(1{)}^{n+1}\to T_{{\mathbb{P}}^n}\to 0$ follows on standard charts by differentiating the ratios $x_i\mathrm{/}{x}_j$: its first map is the coordinate vector, and these derivatives identify the quotient with the tangent bundle. Consequently $\mathrm{td}(T_{{\mathbb{P}}^n})=(h\mathrm{/}(1-e^{-h}){)}^{n+1}$ modulo $h^{n+1}$, so this fibre integral is the formal residue ${\mathrm{Res}}_{h=0}{e}^{mh}(1-e^{-h}{)}^{-n-1}dh$, computed by substituting $z=1-e^{-h}$: then $dh=dz\mathrm{/}(1-z)$ and $e^{mh}=(1-z{)}^{-m}$, and the residue equals the coefficient $[z^n](1-z{)}^{-m-1}=\left(\genfrac{}{}{0px}{}{m+n}{n}\right)$. Comparing with the left-hand side, the identity holds on every generator.

General projective bundles, and conclusion. A rank-zero bundle has empty projectivization and both Riemann-Roch sides are zero; assume positive rank for the following computation. The same computation applies to a general projective bundle projection $\pi :P=\mathbb{P}(\mathcal{E})\to Y$ using the relative projective bundle generators of [L1] and the relative Euler sequence: after splitting $\mathcal{E}$ by a flag bundle the pushforward coefficient of a relative twist is the divided difference ${\sum }_{i}F(-x_i)\mathrm{/}{\prod }_{j\mathrm{\ne }i}(x_j-x_i)$ in the formal roots $x_i$. The relative Euler sequence in the lines convention is $0\to \mathcal{O}\to {\pi }^{\ast }\mathcal{E}\otimes \mathcal{O}(1)\to T_{P\mathrm{/}Y}\to 0$, from first-order deformations of a line. The pushforward formula follows algebraically from the projective relation ${\prod }_i(h+x_i)=0$ and ${\pi }_{\ast }{h}^{r-1}=1$: Lagrange interpolation on the nodes $-x_i$ reads its top remainder coefficient as that divided difference. The generator integrand is $e^{mh}{\prod }_j(h+x_j)\mathrm{/}(1-e^{-x_j-h})$, whose divided-difference sum is the character of the relative symmetric power by the partial-fraction identity ${\prod }_j(1-z{e}^{-x_j}{)}^{-1}={\sum }_i{\prod }_{j\mathrm{\ne }i}(1-e^{x_i-x_j}{)}^{-1}(1-z{e}^{-x_i}{)}^{-1}$; work initially with independent formal roots and invert their differences to justify interpolation and partial fractions. The resulting coefficients of the symmetric-power series are symmetric polynomials, so denominators cancel; truncation in the Chow grading then specializes correctly even when roots coincide. Thus with injective flag pullback and the base Todd projection formula the identity descends. Since the identity is additive in $\alpha$ and holds on the generators of $K_0$ for every $Y$ in the stated hypotheses, the diagram commutes; the case $Y=\mathrm{Spec}k$ is Hirzebruch-Riemann-Roch for ${\mathbb{P}}^n$.