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Let $X$ be a noetherian scheme. Consider the smallest class $\mathcal{S}$ of coherent sheaves on $X$ which has the following closure properties:

  • Every locally free sheaf is in $\mathcal{S}$.
  • If $A,B \in \mathcal{S}$, then $A \otimes B \in \mathcal{S}$.
  • If $C$ is a quotient of $B \in \mathcal{S}$, then $C \in \mathcal{S}$.
  • If $0 \to A \to B \to C \to 0$ is an exact sequence with $A,C \in \mathcal{S}$, then $B \in \mathcal{S}$.

Do we have $\mathcal{S}=\mathsf{Coh}(X)$? If not, is there any more explicit description of the sheaves in $\mathcal{S}$? Of course we have $\mathcal{S}=\mathsf{Coh}(X)$ when $X$ has the resolution property, but I am interested in the general case.

For example let $X$ be the affine plane with a doubled origin. Then every locally free sheaf is trivial. But what is $\mathcal{S}$?

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  • $\begingroup$ If $X$ is regular (for example smooth over a field), then it has the resolution property. $\endgroup$ – answer_bot Mar 7 '14 at 12:13
  • $\begingroup$ I think we also need separated. But anyway, I've removed this remark. $\endgroup$ – Martin Brandenburg Mar 7 '14 at 12:29
  • $\begingroup$ Yes, you are correct. Sorry for the mistake. $\endgroup$ – answer_bot Mar 7 '14 at 13:44
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Suppose you weaken your first condition by only requiring that the structure sheaf $\mathcal{O}_X$ belongs to $\mathcal{S}$. Then the corresponding result does not hold in general.

For instance, you could take $X=\mathbb{P}^1$ and $\mathcal{S}$ to be the class of globaly generated coherent sheaves on $X$, that is sheaves that are finite direct sums of $\mathcal{O}(i)$, $i\geq 0$ and torsion coherent sheaves. You can check by hand that this class satisfies your three last conditions.

For this reason, to answer your question, you have to find a way to construct non-trivial vector bundles on $X$ (and this seems of course difficult).

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    $\begingroup$ Just to add one supporting comment to Olivier's answer: consider the case when $X$ is $\mathbb{P}^n$ with a nontrivial double structure, as in Exercise III.5.9 of Hartshorne's "Algebraic Geometry". How do you know that there is any locally free sheaf other than $\mathcal{O}_X^{\oplus r}$? $\endgroup$ – Jason Starr Mar 7 '14 at 14:27
  • $\begingroup$ "For this reason, to answer your question, you have to find a way to construct non-trivial vector bundles" I don't fully agree. In your example, there are enough vector bundles. Of course there are schemes on which every vector bundle is trivial, but there one has to determine $\mathcal{S}$ then. $\endgroup$ – Martin Brandenburg Mar 7 '14 at 14:33
  • $\begingroup$ @MartinBrandenburg : my point is that it might be expected that $\mathcal{S}=Coh(X)$ under mild hypotheses. If this is the case, your conditions (ii), (iii), (iv) do not prevent you from having to construct vector bundles to prove it. I am sorry if what I meant was not clear in my answer. $\endgroup$ – Olivier Benoist Mar 7 '14 at 15:19

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