## 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

Here

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

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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