= Cohomology under an affine morphism
{title2=$H^i(Y,f_*\mathcal F)\cong H^i(X,\mathcal F)$}
For an <affine morphism> $f:X\to Y$ and a <quasi-coherent sheaf> $\mathcal F$ on $X$, the <direct image sheaf> has the same <sheaf cohomology> on $Y$ as $\mathcal F$ on $X$. To prove this, take a <flasque resolution> $\mathcal F\to\mathcal I^\bullet$. Its direct images are <flasque sheaves>. On each affine open $V\subset Y$, the complex of sections of the direct images is $\Gamma(f^{-1}V,\mathcal I^\bullet)$. By <vanishing of quasi-coherent cohomology on an affine scheme>, this augmented complex is exact. Checking on the affine basis proves that $f_*\mathcal I^\bullet$ resolves $f_*\mathcal F$. Its global sections are identical to those of the original resolution.
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