= Locally free resolution
= Resolution by locally free sheaves
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= Resolutions by locally free sheaves
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A locally free resolution of a <sheaf of modules> $F$ is an <exact sequence> $\cdots\to G_2\to G_1\to G_0\to F\to0$ with each $G_i$ a <locally free sheaf>. If every <coherent sheaf> is the quotient of a <locally free sheaf> of finite rank on a <noetherian scheme>, iteratively taking a <kernel sheaf> constructs such a resolution with <coherent sheaves> at every stage. The <kernel sheaves> need not be <locally free sheaves>, so a finite resolution requires additional hypotheses. A uniform bound on <sheaf cohomology> can nevertheless make a finite initial segment sufficient for <dimension shifting in sheaf cohomology>.
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