Let $X$ be a smooth variety and $G$ a cyclic group acting on it. Assume that the action is free. Then the morphism $\pi: X \to Y=X/G$ is etale. I'm trying to understand why $\pi_\ast \mathcal{O}_X$ decomposes as

$$ \pi_\ast \mathcal{O}_X=\mathcal{O}_Y \oplus L \oplus L^2 \oplus \cdots \oplus L^{n-1} $$

where $n$ is the order of $G$ and $L$ is a line bundle on $Y$ such that $L^n \simeq \mathcal{O}_X$.

Can anybody help me please?

Let $n \geq 2$ be an integer, $X$ a smooth variety over a field $k$ containing $\mu_n$ and $G$ a cyclic group of order $n$ acting on it. Assume that the action is free. Then the morphism $\pi: X \to Y=X/G$ is etale. I'm trying to understand why $\pi_\ast \mathcal{O}_X$ decomposes as

$$ \pi_\ast \mathcal{O}_X=\mathcal{O}_Y \oplus L \oplus L^2 \oplus \cdots \oplus L^{n-1} $$

where $n$ is the order of $G$ and $L$ is a line bundle on $Y$ such that $L^n \simeq \mathcal{O}_X$.

Can anybody help me please?