= Finite twisting resolution of a coherent sheaf on projective space
{title2=$0\to\bigoplus\mathcal O(a_{rj})\to\cdots\to\bigoplus\mathcal O(a_{0j})\to\mathcal F\to0$}
A <coherent sheaf> on $\mathbb P_k^n$ is the <sheaf associated with a graded module> for a finite graded $k[t_0,\ldots,t_n]$-module. Choose a finite graded <free resolution> by the <Hilbert syzygy theorem> and sheafify it; <exactness of localization> preserves exactness. All terms are finite sums of <twisting sheaves on projective space>. Twisting sufficiently positively makes every term acyclic in positive degree, and dimension shifting in <long exact sequences in sheaf cohomology> proves <Serre vanishing>. The graded-module description is supplied by https://stacks.math.columbia.edu/tag/0BXE[Stacks Project, Section 30.15].
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