We consider a quasi-projective noetherian scheme. It is well known that for a coherent sheaf we can construct a sheaf resolution of locally free of finite rank. It is introduced in Hartshorne chapter III.6 for computing Ext and $\mathscr{Ext}$ for example.

In terms of quasi-coherent sheaves, does there always exist a locally free resolution of quasi-coherent sheaves on a quasi-projective noetherian scheme? If exists, what will happen if we do a similar cohomological argument using locally free resolution of infinite rank? If not, how do people talk about this kind of bounded above resolutions in the category of quasi-coherent sheaves? Will we suffer from infinite direct sums?

I know that in general if we remove the condition "quasi-coherent or coherent", we can construct flat resolutions. However, it may be useless in concrete computation for quasi-coherent sheaves.

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    $\begingroup$ Yes, you can find such resolutions: write your quasi-coherent sheaf as an increasing union of coherent sheaves, and then take a gigantic direct sum of locally free sheaves, one hitting each term of the direct system, as the first term of your resolution. This is however unlikely to be very useful unless you can also control what kind of twists show up in the direct sum. $\endgroup$ – Anonymous May 21 '20 at 17:48
  • $\begingroup$ Thank you. There is one thing I am afraid of. Infinite direct sum of quasi-coherent sheaves may not be quasi-coherent. $\endgroup$ – Jiaxi Mo May 22 '20 at 17:18
  • $\begingroup$ Infinite direct sums are fine, it's products that cause problems. $\endgroup$ – Anonymous May 22 '20 at 20:59

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