## 42.66 Grothendieck-Riemann-Roch

Let $(S, \delta )$ be as in Situation 42.7.1. Let $X, Y$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free sheaf ${\mathcal E}$ on $X$ of rank $r$. Let $f : X \to Y$ be a proper smooth morphism. Assume that $R^ if_*\mathcal{E}$ are locally free sheaves on $Y$ of finite rank. The Grothendieck-Riemann-Roch theorem say in this case that

$f_*(Todd(T_{X/Y}) ch(\mathcal{E})) = \sum (-1)^ i ch(R^ if_*\mathcal{E})$

Here

$T_{X/Y} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\Omega _{X/Y}, \mathcal{O}_ X)$

is the relative tangent bundle of $X$ over $Y$. If $Y = \mathop{\mathrm{Spec}}(k)$ where $k$ is a field, then we can restate this as

$\chi (X, \mathcal{E}) = \deg (Todd(T_{X/k}) ch(\mathcal{E}))$

The theorem is more general and becomes easier to prove when formulated in correct generality. We will return to this elsewhere (insert future reference here).

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