= Finiteness of coherent cohomology on projective varieties
For a <coherent sheaf> $\mathcal F$ on a <projective variety> over a field, every $H^q(X,\mathcal F)$ is finite-dimensional, and <Grothendieck vanishing> gives zero for $q>\dim X$. A closed embedding in projective space and a <finite twisting resolution of a coherent sheaf on projective space> reduce finiteness to the explicit <cohomology of twisting sheaves on projective space>. The same resolution proves <Serre vanishing> after sufficiently positive twisting.
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