Fushforward of Line Bundles Between Projective Lines

This is an interesting problem is that for any finite covering $f:\mathbb{P}^1\to \mathbb{P}^1$, how to compute $f_* \mathscr{O}(m)$?

Now we want to find out the following:

Question. Let $f:\mathbb{P}^1\to\mathbb{P}^1$ be a finite morphism of degree $n$, How to compute $f_* \mathscr{O}(m)$ when $m\geq 0$?

Actually there is a proof which the method I saw a long time ago as follows.

Proposition. Let $f:\mathbb{P}^1\to \mathbb{P}^1$ be a finite map of degree $n$ and let $m\geq 0$. Then for $k\geq 0$ if we let \[\ell(m,k)=\mathrm{coeff}_{x^{m-nk}}(1+x+…+x^{n-1})^2,\] then we have \[f_* \mathscr{O}(m)\cong\mathscr{O}(-1)^{\oplus \left(n-\sum_{k\geq 0}\ell(m,k)\right)}\oplus\bigoplus_{k\geq 0}\mathscr{O}(k)^{\oplus \sum_{k\geq 0}\ell(m,k)}.\]

Proof. We know that $f$ is flat automatically since modules over some Dedekind ring is flat if and only if it is torsion free (see Tag 0AUW). Then $f_* \mathscr{O}(m)$ is a vector bundle of rank $n$ over $\mathbb{P}^1$. By the classification of vector bundles we can let

By projection formula, we have $f_* (\mathscr{O}(m-nk))\cong f_* (\mathscr{O}(m))\otimes\mathscr{O}(-k)$, hence we get $\ell(m,k)=\ell(m-nk,0)$. Taking $h^0$ at (3) we have $m+1=\sum_{k\geq 0}(k+1)\ell(m,k)$. Hence we have

\begin{align*} \sum_{m\geq 0}(m+1)x^m&=\sum_{m\geq 0}\sum_{k\geq 0}\ell(m,k)(k+1)x^m\\ &=\sum_{m\geq 0}\sum_{k\geq 0}(k+1)x^{nk}\ell(m,k)x^{m-nk}\\ &=\sum_{k\geq 0}(k+1)x^{nk}\sum_{m\geq 0}\ell(m,k)x^{m-nk}\\ &=\sum_{k\geq 0}(k+1)x^{nk}\sum_{m\geq 0}\ell(m-nk,0)x^{m-nk}\\ &=\sum_{k\geq 0}(k+1)x^{nk}\sum_{m\geq 0}\ell(m,0)x^{m} \end{align*}

where the last equation holds because for any $j<0$ we have $\ell(j,0)=0$ by checking global sections and then $\sum_{m\geq 0}\ell(m-nk,0)x^{m-nk}$ will be a constant when $k$ varies.

Hence by taking $m$ to be $m-nk$, hence when $k\geq 0$ we have

\begin{align*} \ell(m,k)&=\mathrm{coeff}_{x^{m-nk}}\frac{\sum_{m\geq 0}(m+1)x^m}{\sum_{k\geq 0}(k+1)x^{nk}}\\ &=\mathrm{coeff}_{x^{m-nk}}\left(\frac{1-x^n}{1-x}\right)^2\\ &=\mathrm{coeff}_{x^{m-nk}}(1+x+...+x^{n-1})^2. \end{align*}

Now we consider the case when $k<0$. Actually when $k\leq -2$ we get $h^1(\mathbb{P}^1,\mathscr{O}(k))>0$. Hence taking $h^1$ at (3) again we get $\ell(m,k)=0$ in this case! Hence we let

and we get the solution! $\blacksquare$

Corollary. In this case we have $f_* \mathscr{O}\cong \mathscr{O}\oplus\mathscr{O}(-1)^{\oplus(n-1)}$ and $f_* \mathscr{O}(1)\cong \mathscr{O}^{\oplus 2}\oplus\mathscr{O}(-1)^{\oplus(n-2)}$.

Similarly you can use the same method to compute other similar cases.