= Dimension shifting in sheaf cohomology
{title2=$H^q(X,K_j)\cong H^{q+1}(X,K_{j+1})$}
From a <short exact sequence of sheaves> $0\to K_{j+1}\to G_j\to K_j\to0$, the <long exact sequence in sheaf cohomology> gives the displayed <isomorphism> whenever $H^q(X,G_j)=H^{q+1}(X,G_j)=0$. Repeating it along a <locally free resolution> shifts a fixed group to successively higher degrees. If all relevant <kernel sheaves> are <quasi-coherent sheaves> and there is a uniform vanishing bound for their <sheaf cohomology>, sufficiently many shifts force the original group to vanish. The <cohomological dimension bound from an affine cover> supplies such a bound for a <closed subscheme> of <projective space>.
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