Let $\mathcal F$ be a locally free sheaf over a scheme or variety $X$. Then we know that $\mathcal F$ can be thought of as sections to a certain vector bundle $E \to X$. Suppose more generally that $\mathcal F$ is only a coherent sheaf, over a nice space like a variety. Can $\mathcal F$ be viewed as sections of some space $E \to X$? Thank you.

  • $\begingroup$ Every sheaf is sections of its etale space. $\endgroup$ – ziggurism Sep 11 '17 at 22:57

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