Pushforward of invertible sheaf on $P^1$ by finite flat map Let $k$ be an algebraically closed field. Any finite $k$-morphism $P^1_k\rightarrow P^1_k$ is flat (miracle flatness) and surjective on the underlying spaces. Therefore, the pushforward of a coherent locally free sheaf is coherent locally free (on $P^1_k$, such sheaves can be described by a finite sequence of integers using the fact that Picard rank is 1 and there is Birkhoff--Grothendieck splitting). 
Assume we have a finite $k$-morphism $P^1_k\rightarrow P^1_k$ such that the inverse image of the generic point has cardinality $n\geq 2$. Which sheaves can we get as the pushforward of a locally free sheaf of rank 1? 
 A: Write $$ f_* \mathcal{O}(m) = \bigoplus_{k\in\mathbb{Z}} \mathcal{O}(k)^{\alpha(m, k)}.  $$ We want to compute the multiplicities $\alpha(m,k)$. We have $f^* \mathcal{O}(k) = \mathcal{O}(nk)$ where $n = \deg f$, so the projection formula gives $$ (f_* \mathcal{O}(m)) \otimes \mathcal{O}(-k) = f_* \mathcal{O}(m-nk),$$ 
and hence $\alpha(m,k) = \alpha(m-nk, 0)$. If $S(x) = \sum_m (m+1)x^m = 1/(1-x)^2$ and $A(x) = \alpha(m,0) x^m$, then applying $h^0(-)$ to both sides of the first displayed formula, multiplying by $x^m$ and summing over $m\in\mathbb{Z}$ gives $$ S(x) = A(x)\cdot S(x^n).$$
Thus 
$$ \alpha(m,k) = \left(\text{coefficient of }x^{m-nk}\text{ in } S(x)/S(x^n) = (1+x+\cdots + x^{n-1})^2\right). $$
For example, $$f_* \mathcal{O} = \mathcal{O} \oplus \mathcal{O}(-1)^{n-1}.$$
(The above argument appears in my paper "Frobenius Push-Forwards on Quadrics", and works similarly for $\mathbb{P}^N$. The first place I know where these pushforwards are computed is the paper "Frobenius direct images of line bundles on toric varieties" by J. F. Thomsen)
A: Here is another way to view this.
Let's look at local sections of $\mathbb{P}^1_w$. Cover $\mathbb{P}^1_w$ by affine charts $U_0$ and $U_1$. Then $\mathcal{O}_{\mathbb{P}^1_w}$ and $\mathcal{O}_{\mathbb{P}^1_w}(-1)$ can be viewed as following pictures:
$$
 \rlap{\underbrace{\phantom{\cdots w^{-3}, w^{-2}, w^{-1}}}_{U_0}} w^{-3}, w^{-2}, w^{-1}, 
     \overbrace{1, w , w^2, w^3, \cdots}^{U_1} 
$$
$$
\underbrace{\cdots w^{-3}, w^{-2}, w^{-1}, 1\,}_{U_0} , \, 
\overbrace{ w , w^2, w^3, \cdots}^{U_1}
$$
For example, the local sections of $\mathcal{O}_{\mathbb{P}^1_w}$ over Spec$k[w]$ are $1, w, w^2,\cdots$ and over Spec$k[w^{-1}]$ are $1, w^{-1}, w^{-2},\cdots$. So the global section is only $k$.
Now suppose we have the morphism given by $z=w^3$. Then we can decompose $\mathcal{O}_{\mathbb{P}^1_w}$ as following three:
$$
 \rlap{\underbrace{\phantom{\cdots w^{-9}, w^{-6}, w^{-3}}}_{U_0}} w^{-9}, w^{-6}, w^{-3}, 
     \overbrace{1, w^3 , w^6, w^9, \cdots}^{U_1} 
$$
$$
\underbrace{\cdots w^{-8}, w^{-5}, w^{-2}}_{U_0}, \,
\overbrace{ w^{1}, w^4 , w^7, \cdots}^{U_1}
$$
$$
\underbrace{\cdots w^{-7}, w^{-4}, w^{-1}}_{U_0}, \,
\overbrace{ w^{2}, w^5 , w^8, \cdots}^{U_1}
$$
Hence
$$f_* \mathcal{O}_{\mathbb{P}^1_w} = \mathcal{O}_{\mathbb{P}^1_z} \oplus \mathcal{O}_{\mathbb{P}^1_z}(-1)^{2}.$$
