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I hope I wasn't searching wrong keywords or overlooking some easy arguments to prove/disprove it. What I'm asking is the following:

Let $X$ be a smooth complete toric variety. $\mathcal F$ a coherent sheaf (or a torus equivariant one if the previous one does not work). Does $\mathcal F$ always admit a finite resolution by sums of line bundles, i.e., a resolution of $\mathcal F$ that looks like

$$0\rightarrow\bigoplus_{i=1}^{a_n}L_{i,n}\rightarrow\dotsb\rightarrow \bigoplus_{i=1}^{a_0}L_{i,0}\rightarrow\mathcal F\rightarrow 0$$

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  • $\begingroup$ This is definitely true if $X$ has a full exceptional collection consisting of line bundles. This happens quite often for small dimension or small Picard rank, however not always. $\endgroup$
    – Sasha
    Commented Jan 20, 2015 at 8:26
  • $\begingroup$ @Sasha: Do you have a counter-example (with $X$ a smooth toric variety)? $\endgroup$
    – abx
    Commented Jan 20, 2015 at 10:21
  • $\begingroup$ See Efimov: arXiv:1010.3755 for a smooth toric Fano variety without a full, strong exceptional collection of line bundles. $\endgroup$
    – Richard Eager
    Commented Jan 20, 2015 at 13:50
  • $\begingroup$ @Sasha If you don't mind, could you tell me how to get the resolution from a full exceptional collection of line bundles? Thanks very much. $\endgroup$
    – Honglu
    Commented Jan 20, 2015 at 23:33
  • $\begingroup$ @abx: I had in mind the paper of Efimov which was mentioned by Richard in his comment. $\endgroup$
    – Sasha
    Commented Jan 21, 2015 at 1:18

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