Riemann-Roch for regular embeddings

Model case. By finite locally free resolution and additivity, it suffices to prove the formula for a vector bundle $E$; assume this through the deformation argument. Let $P={\mathbb{P}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}}(N\oplus 1)$ with projection $p:P\to X$ and universal sequence $0\to \mathcal{O}(-1)\to p^{\ast }(N\oplus 1)\to Q\to 0$, so that the image of the summand $1$ in $Q$ gives a section of $Q$ whose zero scheme is the distinguished section $s:X\hookrightarrow P$, a regular embedding of codimension $d$ and $s^{\ast }Q=N$. To check the section hypothesis of [L1], trivialize $N$ over $\mathrm{Spec}A$ and use the chart containing $s(X)$ where the last coordinate of the tautological line is $1$: its generator is $(z_1,\dots ,z_d,1)$, and $Q$ has basis the images of the first $d$ standard vectors. The image of the last standard vector has components $-z_1,\dots ,-z_d$ in this basis. Each is a nonzerodivisor after the preceding coordinates are killed in $A[z_1,\dots ,z_d]$, giving a regular sequence of length $d$, including the empty sequence when $d=0$. Away from $s(X)$ some component is a unit locally, since the line is not the last summand. The Koszul resolution of the regular section of [L1], tensored with $p^{\ast }E$, gives $\mathrm{ch}(s_{\ast }E)=\mathrm{ch}(p^{\ast }E){\sum }_{j=0}^d(-1{)}^j\mathrm{ch}({\bigwedge }^j{Q}^{\vee })$ in $A^{\ast }(P{)}_{\mathbb{Q}}$: after injective flag pullback, if $x_1,\dots ,x_d$ are the Chern roots of $Q$, the alternating sum of exterior powers is ${\prod }_j(1-e^{-x_j})=c_d(Q)\mathrm{td}(Q{)}^{-1}$ by [L3]. The regular-section formula gives $c_d(Q)=s_{\ast }[X]$ and the ring projection formula gives $c_d(Q)\gamma =s_{\ast }(s^{\ast }\gamma )$ for every $\gamma \in A^{\ast }(P)$; substituting $\gamma =\mathrm{td}(Q{)}^{-1}\mathrm{ch}(p^{\ast }E)$ gives $\mathrm{ch}(s_{\ast }E)=s_{\ast }(\mathrm{td}(N{)}^{-1}\mathrm{ch}(E))$, the asserted formula in the model case.

Reduction to the model. Let $b:M\to Y$ be the deformation blowup of [L2] with $F:X\times {\mathbb{P}}^1\hookrightarrow M$ the strict transform, $j_0:Y\hookrightarrow M$ the ordinary fibre, $k:P\hookrightarrow M$ the fibre $P=\mathbb{P}(N\oplus 1)$ and $l:B\hookrightarrow M$ the strict transform of $Y\times \{\mathrm{\infty }\}$. Resolve the sheaf $F_{\ast }{\mathrm{pr}}_X^{\ast }E$ by a bounded complex $G_{\bullet }$ of vector bundles and put $\gamma ={\sum }_j(-1{)}^j\mathrm{ch}(G_j)$, an operational class on $M$. By [L1] this complex restricts on $j_0$ to a resolution of $i_{\ast }E$, on $k$ to a resolution of $s_{\ast }E$, and is exact on $l$. Their Chern characters restrict along these smooth embeddings because they are polynomials in Chern classes, whose naturality holds for arbitrary smooth-scheme morphisms. The two Cartier fibres at $0$ and $\mathrm{\infty }$ are linearly equivalent, so the cycle identity $j_{0\ast }[Y]=k_{\ast }[P]+l_{\ast }[B]$ holds in $A_{\ast }(M)$ with multiplicity one because the centre is smooth; capping with $\gamma$ and using the Chern projection formula gives $j_{0\ast }(\mathrm{ch}(i_{\ast }E)\cap [Y])=k_{\ast }(\mathrm{ch}(s_{\ast }E)\cap [P])$, the $B$-term vanishing because the restricted complex is exact.

Conclusion. Push the equality of step 1.2 forward along the proper morphism $b$: since $b{j}_0={\mathrm{id}}_Y$ and $bk=ip$, and proper pushforward is functorial with the projection formula [L4], one gets $\mathrm{ch}(i_{\ast }E)\cap [Y]=i_{\ast }{p}_{\ast }(\mathrm{ch}(s_{\ast }E)\cap [P])=i_{\ast }(\mathrm{ch}(E)\mathrm{td}(N{)}^{-1}\cap [X])$ by the model computation of step 1.1 and $ps={\mathrm{id}}_X$. Hence $\mathrm{ch}(i_{\ast }E)=i_{\ast }(\mathrm{ch}(E)\mathrm{td}(N{)}^{-1})$; the equivalent Todd-tangent form follows from the exact tangent sequence $0\to T_X\to i^{\ast }{T}_Y\to N\to 0$ and the multiplicativity of the Todd class, using $[i^{\ast }{T}_Y]=[T_X]+[N]$ and no bundle splitting. The finite locally free resolution extends the identity from vector bundles to all classes of $K_0(X)$ by additivity.