We all know that smooth morphisms have sections etale locally. However, the following similar statement is not obvious for me:
If X->Y->Z, X is etale over Y, Y is finite and surjective over Z, then a section of X->Y exists etale locally on Z, i.e. there exists an etale cover U of Z such that X_U->Y_U has a section. Where _U means pullback on U.
I think it is supposed to be easy.
Can anyone explain this to me? Thanks.