Euler characteristic for pull back of a coherent sheaf under finite etale covering

Question

Let $f: X \rightarrow Y$ be a finite etale morphism between smooth projective varieties over $\mathbb C$ with degree $n>0$. Given a coherent sheaf $\mathcal{F}$ on $Y$, do we necessarily have $\chi(X,f^*\mathcal{F})=n \chi(Y,\mathcal{F})$ (if we know this hold for $O_Y$) ? Maybe the Leray spectral sequence is helpful.

Answer 1

The formula $\chi(X, f^*(\mathcal{F}))=n \chi (Y, \mathcal{F})$ is true:

Let $\text{dim}(Y)=N$. For any $\alpha \in H^{2N}(Y,\mathbb{Q})$, we have $$\text{deg}(f^*\alpha) = n\text{deg}(\alpha)$$

Since $f$ is etale, we also have an isomorphism of tangent bundles $f^*\mathcal{T}_Y \cong \mathcal{T}_X$. Applying these facts in the case $$\alpha = [\text{Ch}(\mathcal{F}) \cdot \text{td}({\mathcal{T}_Y})]_{2N}$$ (where $\text{Ch}$ is the Chern character, $\text{td}$ is the Todd class, and $\beta_i$ is the degree-$i$ part of $\beta \in H^*(Y,\mathbb{Q})$) and applying the Hirzebruch-Riemann-Roch formula, we obtain:

\begin{align} \chi(X, f^*(\mathcal{F})) \\ &=\text{deg}([\text{Ch}(f^*\mathcal{F}) \cdot \text{td}({\mathcal{T}_X})]_{2N}) \\ &=\text{deg}([\text{Ch}(f^*\mathcal{F}) \cdot \text{td}(f^*\mathcal{T}_Y)]_{2N}) \\ &=\text{deg}([f^*\text{Ch}(\mathcal{F}) \cdot f^*\text{td}(\mathcal{T}_Y)]_{2N}) \\ &=n\text{deg}([\text{Ch}(\mathcal{F}) \cdot \text{td}(\mathcal{T}_Y)]_{2N}) \\ &=n \chi (Y, \mathcal{F}) \end{align}