I was reading the note on the proof of Borel-Weil-Bott theorem written by Jacob Lurie (See http://www.math.harvard.edu/~lurie/papers/bwb.pdf) and I was stuck at the point where he said that for a $\mathbb{P}^1$-bundle $\pi: E \longrightarrow S$ and any line bundle ${\cal L}$ on $E$ with fibre degree $n$, we have, by descent theory, the following isomorphism: $$\pi_*{\cal L}\cong R^1\pi_*({\cal L}\otimes K^{\otimes n+1})~~(*)$$ where $K$ is according to him the relative canonical bundle. It is clear to me that the sheaves on both sides are vector bundles over $S$ with fibres $H^0(\mathbb{P}^1,{\cal L}|_{\mathbb{P}^1})$ and $H^1(\mathbb{P}^1,{\cal L}|_{\mathbb{P}^1}\otimes K_{\mathbb{P}^1}^{\otimes n+1} )$ respectively which are (non-canonically) isomorphic by Serre duality. Notice this holds whenever we have two line bundles whose fibre degrees sum to $-2$ but I don't believe the isomorphism (*) holds for any such pair of line bundles.
My question is what is the assumption required in order to make (*) hold? What content of descent theory did he use? I am also confused why Serre duality takes this form instead of the usual one ${\cal L}^{\vee}\otimes K$. Can we get the similar conclusion based on this usual form, i.e. $$\pi_*{\cal L}\cong R^1\pi_*({\cal L}^{\vee}\otimes K)?$$