Dimension shifting in sheaf cohomology ($H^q(X,K_j)\cong H^{q+1}(X,K_{j+1})$)

From a short exact sequence of sheaves $0\to K_{j+1}\to G_j\to K_j\to 0$, 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.