# Algebraic Geometry – To prove the theorem of Borel-Weil-Bott: Two derived forward thrust slices are isomorphic after the descent theory

I read the note on the proof of the theorem of Borel Weil Bott by Jacob Lurie (see http://www.math.harvard.edu/~lurie/papers/bwb.pdf), and I stuck to the spot he stuck said that for one $$mathbb {P} ^ 1$$-bundle up $$pi: E longrightarrow S$$ and any trunk group $${ cal L}$$ on $$E$$ with fiber grade $$n$$According to the theory of descent, we have the following isomorphism:
$$pi _ * { cal L} cong R ^ 1 pi _ * ({ cal L} times K ^ { times n + 1}) ~~ (*)$$
from where $$K$$ after him is the relative canonical bundle.
I realize that the slices on both sides are vector bundles $$S$$ with fibers $$H ^ 0 ( mathbb {P} ^ 1, { cal L} | _ { mathbb {P} ^ 1})$$ and $$H ^ 1 ( mathbb {P} ^ 1, { cal L} | _ { mathbb {P} ^ 1} otimes K _ { mathbb {P} ^ 1} ^ { otimes n + 1} )$$ They are (not canonical) isomorphic to Serre Duality. Note that this applies if we have two line bundles whose fiber degrees add up $$-2$$ However, I do not believe that the isomorphism (*) holds for such a pair of wires.

My question is, what is the assumption to be able to hold (*)? What content of descent did he use? I am also confused as to why Serre Duality takes this form, rather than the usual one $${ cal L} ^ { vee} times K$$, Can we get the similar conclusion on the basis of this usual form, d. H.
$$pi _ * { cal L} cong R ^ 1 pi _ * ({ cal L} ^ { vee} otimes K)?$$