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I am trying to get a better understanding of quasicoherent sheaves and was able to construct examples and counterexamples for affine schemes $X = Spec(R)$. Like most people I've talked to, I chose R to be a DVR for doing that. But now I don't seem to be able to construct counterexamples if the scheme is the projective space $X = \mathbb{P}^n_k$.

Is there an "easy" way to construct non-quasicoherent $\mathcal{O}_X$-modules? Maybe just for small $n \geq 1$?

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  • $\begingroup$ Simple method to apply what you already know: find a subscheme of $X$ that is isomorphic to the spectrum of a DVR. $\endgroup$ – Hurkyl Oct 31 '17 at 7:51
  • $\begingroup$ What you could do for example, is take the skycraper sheaf on the stalk $\mathcal{O}_{X,x}$. $\endgroup$ – user45878 Oct 31 '17 at 14:14

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