In http://arxiv.org/abs/1210.2564 Example 4.12 it is written that for $Y$ the blowup of $\mathbb A^2$ at the origin (i.e. $Y \cong \mathrm{Tot} \, \mathcal O_{\mathbb P^1} (-1)$), and $π \colon Y \to \mathbb P^1$ the projection, we have
$$ π_∗ (\mathcal O_Y) \cong \bigoplus_{p≤0} \mathcal O_{\mathbb P^1} (−1)^{⊗p} = \bigoplus_{k≥0} \mathcal O_{\mathbb P^1} (k) $$
I don't understand how to derive the first isomorphism.
More generally, given any other coherent sheaf $\mathcal F$ on $Y$ (or on any other scheme/variety $X$ with an affine morphism $X \to \mathbb P^1$) is there a general method of finding its pushforward under $π$?