Grothendieck–Riemann–Roch theorem

Remarks

The idea of the proof: factor $f$ as a closed immersion $X \to \mathbb{P}^N_Y$ followed by the projection $\mathbb{P}^N_Y \to Y$. For the projection, both sides are computed from the explicit structure of $K_0$ and of the Chow ring of a projective bundle. For a closed immersion, one deforms to the normal bundle and reduces to the zero section of a vector bundle, where the Koszul complex gives the formula.

Example: for $Y = \operatorname{Spec} k$, $A^*(Y)_\Q = \Q$, $\operatorname{td}(T_Y) = 1$ and $\operatorname{ch}(f_! [\mathcal{F}]) = \chi(X, \mathcal{F})$. So the theorem becomes the Hirzebruch–Riemann–Roch formula $\chi(X, \mathcal{F}) = \int_X \operatorname{ch}(\mathcal{F}) \operatorname{td}(T_X)$.

Without the Todd classes the formula is false: the Chern character alone does not commute with pushforward, as already the case $f \colon \mathbb{P}^1_k \to \operatorname{Spec} k$ with $\mathcal{F} = \mathcal{O}$ shows, where $\chi = 1$ but $f_* \operatorname{ch}(\mathcal{O}) = 0$.

Going deeper

For a student

The theorem says that the Chern character, corrected by Todd classes, is compatible with pushforward: a commutative square between K-theory and the Chow ring. It is used to compute Chern classes of Hodge bundles and other natural bundles on moduli spaces, such as Mumford's computation on the moduli space of curves. The case of a point recovers the Hirzebruch–Riemann–Roch theorem, and the topological analogue is the Atiyah–Singer family index theorem.