Euler character of etale finite cover Let $\pi: \tilde{X} \to X$ be an etale finite cover, then why the Euler character has relation:
$$\chi(\tilde{X},\mathcal{O}_{\tilde{X}})=\deg(\pi)\chi({X},\mathcal{O}_{{X}}).$$
I try to use  Riemann-Roch, but do not know how to relate Chern characters and Todd class of them. 
Besides, I found a similar question on topological setting.
 A: Because $\pi$ is étale, you have $\pi^\ast(\mathcal T_X)=\mathcal T_{\tilde X}$ and $\pi^\ast(\mathcal O_X)=\mathcal O_{\tilde X}$ anyway. Using Hirzebruch-Riemann-Roch and the below Lemma,
we get
\begin{align*}
\deg(\pi)\cdot \chi(X,\mathcal O_X)
&=\deg(\pi)\cdot\int_X \Bigl(\mathrm{ch}(\mathcal O_X)\cdot\mathrm{td}(\mathcal T_X)\Bigr)
= \int_{\tilde X} \pi^\ast\Bigl(\mathrm{ch}(\mathcal O_X)\cdot\mathrm{td}(\mathcal T_X)\Bigr) 
\\ &= \int_{\tilde X} \mathrm{ch}(\mathcal O_{\tilde X})\cdot\mathrm{td}(\mathcal T_{\tilde X}) = \chi(\tilde X,\mathcal O_{\tilde X}).
\end{align*}
Lemma Let $\pi:\tilde X\to X$ be a finite surjective morphism of nonsingular varieties of dimension $n$. Denote by $A(X)$ the Chow ring of $X$. The composite
$$A(X)\xrightarrow{\quad\textstyle\pi^\ast\quad} A(\tilde X)\xrightarrow{\quad\textstyle\pi_\ast\quad} A(X)$$
is multiplication by $N=\deg(\pi)$. In particular, for all $\alpha\in A^n(X)$,
$$\int_{\tilde X} \pi^\ast(\alpha) = N\cdot\int_X \alpha$$
Proof This is Example 1.7.4 from  Fulton's book Intersection Theory, but I can give a really short proof using Formula 12.6.2 from Görz-Wedhorn if we are dealing with varieties and the morphism is étale. By said formula, for any point $P\in X$, we have $N=\sum_{\pi(Q)=P} [\Bbbk(Q):\Bbbk(P)]$ because the ramification indices are all $1$. Hence,
$$N\cdot\int_X [P] = N\cdot[\Bbbk(P):\Bbbk]=\sum_{\pi(Q)=P} [\Bbbk(Q):\Bbbk(P)][\Bbbk(P):\Bbbk] = \sum_{\pi(Q)=P} [\Bbbk(Q):\Bbbk]  = \int_{\tilde X} \pi^\ast[P].$$
We are done because this extends to formal sums of points.
Comment 1: The Lemma does not require the étale condition because points can effortlessly be moved out of the ramification locus. I left this out because, well, we don't have ramification.
Comment 2: The more I look at this, the more I think that this can probably be generalized to quasi-projective schemes over fields, using Grothendieck-Riemann-Roch. I am not sure how general you would like your objects $X$ and $\tilde X$ to be.
A: This follows from Riemann-Hurwitz, and from the fact that an étale cover is unramified.
