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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.

Fix a coherent sheaf $F$ and an integer $k>M$. Set $K_0=F$. Repeatedly apply a to construct short exact sequences of sheaveswhere each $G_j$ is a coherent sheaf and a locally free sheaf. Each kernel sheaf $K_{j+1}$ is a coherent sheaf because $X$ is a noetherian scheme. Thus the construction can be repeated as many times as needed; it does not require the kernel sheaves themselves to be locally free sheaves.

For every integer $q>M$, the long exact sequence in sheaf cohomology containsBoth outer groups vanish by the hypothesis, since $q$ and $q+1$ exceed $M$. Hence ${\delta }_j$ is an isomorphism. Iterating this dimension shifting in sheaf cohomology gives

Choose $s\ge 0$ with $k+s>N$, where $N$ is the uniform bound from b. The coherent sheaf $K_s$ is a quasi-coherent sheaf, so b makes the last group zero. ThereforeThe argument uses a finite initial segment of a locally free resolution. There is no need to assume that every coherent sheaf has a finite locally free resolution: the uniform bound from b is what terminates the argument.