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By construction the sheaf $\pi_* \mathcal{O}_X$ is a $G$-equivariant vector bundle of rank $n$ on $Y$ and, since $G$ is a cyclic group, the representation of $\pi_* \mathcal{O}_X$ as a $G$-module splits into linear factors.

Now take any isomorphism $G \cong \mathbb{Z}/ n \mathbb{Z}$, and call $L$ the eigensheaf of $\pi_* \mathcal{O}_X$ corresponding to the generator $\bar{1} \in G$. Clearly $L$ is a $n$-torsion line bundle on $X$ and moreover for any $k \in \mathbb Z$ the eigensheaf corresponding to $\bar{k} \in G$ is precisely $L^k$.

So you obtain the desired splitting.

By construction the sheaf $\pi_* \mathcal{O}_X$ is a $G$-equivariant vector bundle of rank $n$ on $Y$ and, since $G$ is a cyclic group, the representation of $\pi_* \mathcal{O}_X$ as a $G$-module splits into direct summands which are all line bundles.

Now take any isomorphism $G \cong \mathbb{Z}/ n \mathbb{Z}$, and call $L$ the eigensheaf of $\pi_* \mathcal{O}_X$ corresponding to the generator $\bar{1} \in G$. Clearly $L$ is a $n$-torsion line bundle on $X$ and moreover for any $k \in \mathbb Z$ the eigensheaf corresponding to $\bar{k} \in G$ is precisely $L^k$.

So you obtain the desired splitting.

By construction the sheaf $\pi_* \mathcal{O}_X$ is a $G$-equivariant vector bundle of rank $n$ on $Y$ and, since $G$ is a cyclic group, the representation of $\pi_* \mathcal{O}_X$ as a $G$-module splits into linear factors.

Now take any isomorphism $G \cong \mathbb{Z}/ n \mathbb{Z}$, and call $L$ the eigensheaf corresponding to the generator $\bar{1} \in G$. Clearly $L$ is a $n$-torsion line bundle on $X$ and moreover for any $k \in \mathbb Z$ the eigensheaf of $\pi_* \mathcal{O}_X$ corresponding to $\bar{k} \in G$ is precisely $L^k$.

So you obtain the desired splitting.

By construction the sheaf $\pi_* \mathcal{O}_X$ is a $G$-equivariant vector bundle of rank $n$ on $Y$ and, since $G$ is a cyclic group, the representation of $\pi_* \mathcal{O}_X$ as a $G$-module splits into linear factors.

Now take any isomorphism $G \cong \mathbb{Z}/ n \mathbb{Z}$, and call $L$ the eigensheaf of $\pi_* \mathcal{O}_X$ corresponding to the generator $\bar{1} \in G$. Clearly $L$ is a $n$-torsion line bundle on $X$ and moreover for any $k \in \mathbb Z$ the eigensheaf corresponding to $\bar{k} \in G$ is precisely $L^k$.

So you obtain the desired splitting.

By construction the sheaf $\pi_* \mathcal{O}_X$ is a $G$-equivariant vector bundle of rank $n$ on $Y$ and, since $G$ is a cyclic group, the representation of $\pi_* \mathcal{O}_X$ as a $G$-module splits into linear factors.

Now take any isomorphism $G \cong \mathbb{Z}/ n \mathbb{Z}$, and call $L$ the eigensheaf corresponding to the natural generator $\bar{1} \in G$. Clearly $L$ is a $n$-torsion line bundle on $X$ and moreover for any $k \in \mathbb Z$ the eigensheaf of $\pi_* \mathcal{O}_X$ corresponding to $\bar{k} \in G$ is precisely $L^k$.

So you obtain the desired splitting.

By construction the sheaf $\pi_* \mathcal{O}_X$ is a $G$-equivariant vector bundle of rank $n$ on $Y$ and, since $G$ is a cyclic group, the representation of $\pi_* \mathcal{O}_X$ as a $G$-module splits into linear factors.

Now take any isomorphism $G \cong \mathbb{Z}/ n \mathbb{Z}$, and call $L$ the eigensheaf corresponding to the generator $\bar{1} \in G$. Clearly $L$ is a $n$-torsion line bundle on $X$ and moreover for any $k \in \mathbb Z$ the eigensheaf of $\pi_* \mathcal{O}_X$ corresponding to $\bar{k} \in G$ is precisely $L^k$.

So you obtain the desired splitting.