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Suppose $X$ is an algebraic variety, is there a coherent sheaf $\mathcal{F}$ on $X$ which is not a quotient of locally free sheaf?

(Hartshorne II Cor 5.18 showed that on every projective variety, coherent sheaves are quotient of locally free sheaves of finite rank, by extension of coherent sheaves, the same holds for quasi-projective varieties.)

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There are smooth schemes which don't have this property, usually called resolution property. A simple example is $X=\mathbb{A}^n \cup_{\mathbb{A}^n \setminus \{0\}} \mathbb{A}^n$, the affine $n$-space with a double origin, and $n \geq 2$. A locally free sheaf on $X$ pulls back to two locally free sheaves on $\mathbb{A}^n$ which become isomorphic when restricted to $\mathbb{A}^n \setminus \{0\}$. Since every locally free sheaf on $\mathbb{A}^n$ is free (Quillen-Suslin Theorem), the isomorphism corresponds to an invertible matrix of global sections of $\mathbb{A}^n \setminus \{0\}$, which by Hartog's Lemma coincide with the global sections of $\mathbb{A}^n$ (here we use $n \geq 2$). Hence, the isomorphism of the two free sheaves extends on $\mathbb{A}^n$, which means that the original locally free sheaf on $X$ is free. It follows that not every coherent sheaf on $X$ is a quotient of a locally free i.e. free sheaf, because otherwise $\mathcal{O}_X$ would be ample and therefore $X$ would be separated.

On the other hand, many schemes have the (strong) resolution property, for example divisorial schemes (M. Borelli, Divisorial varieties) - including projective schemes and any separated noetherian locally factorial scheme (SGA 6, Exp. II, Proposition 2.2.7), as well as any separated algebraic surface (P. Gross. The resolution property of algebraic surfaces). For further results see (P. Gross, Vector bundles as generators on schemes and stacks. PhD thesis) and (B. Totaro, The resolution property for schemes and stacks). For algebraic stacks, there is a useful criterion which depends on a presentation (D. Schäppi, A characterization of categories of coherent sheaves of certain algebraic stacks).

It seems to be an open problem if there is a variety (separated scheme of finite type) which doesn't have the resolution property.

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  • $\begingroup$ Very nice answer. $\endgroup$ – Mariano Suárez-Álvarez Jun 28 '14 at 7:28
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    $\begingroup$ Certainly a great answer. The definition of variety varies from source to source, but is there any well-known source that allows a variety to not be separated? $\endgroup$ – RghtHndSd Jun 28 '14 at 15:30
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    $\begingroup$ You are right (only in Liu's book varieties may be everything of finite type). Totaro proves that every scheme with the resolution property is semi-separated (i.e. has affine diagonal). D. Schäppi told me that there is no known example of a semi-separated scheme which doesn't have the resolution property ... see also mathoverflow.net/questions/25122 $\endgroup$ – Martin Brandenburg Jun 28 '14 at 15:40
  • $\begingroup$ Actually Totaro's Question 1 in his paper asks if every (semi) separated scheme has the resolution property ... it may be an open question. $\endgroup$ – Martin Brandenburg Jun 28 '14 at 16:07
  • $\begingroup$ Sorry I have a small question: If every coherent sheaf on $X$ is a quotient of a locally free, how does it imply $\mathcal{O}_X$ be ample? Thanks! $\endgroup$ – Qixiao Jun 29 '14 at 6:24

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