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Homotopy theory, homological algebra, algebraic treatments of manifolds.
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Sheaf cohomology and double covers
Hints: I would show that $R^i\pi_*O_X=0$ for $i>0$, and then I would use adjunction.
(2) holds in general. Basically, you need to show that $\Gamma(Y,A\oplus B)=\Gamma(Y,A)\oplus \Gamma(Y,B)$.