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A quasi compact and quasi-separated scheme has its derived category of sheaves of modules with quasi-coherent cohomology generated by perfect complexes. This is actually a theorem of Bondal and Van de Berg Generators and representability of functors in commutative and noncommutative geometry. However, there are non-separated schemes that, despite the previous theorem, do not posses the resolution property, see the paper by Totaro The resolution property for schemes and stacks for a discussion of the subtleties of the issue.

The proof by Bondal-Van den Bergh uses Thomason-Neeman localization: a perfect complex overt a dense open subset extends. This does not hold for vector bundles in general. This is where the difference between compact generation and generation by vector bundles lies.

A quasi compact and quasi-separated scheme has its derived category of sheaves of modules with quasi-coherent cohomology generated by perfect complexes. This is actually a theorem of Bondal and Van de Berg Generators and representability of functors in commutative and noncommutative geometry. However, there are non-separated schemes that, despite the previous theorem, do not posses the resolution property, see the paper by Totaro The resolution property for schemes and stacks for a discussion of the subtleties of the issue.

The proof by Bondal-Van den Bergh uses Thomason-Neeman localization: a perfect complex over a dense open subset extends. This does not hold for vector bundles in general. This is where the difference between compact generation and generation by vector bundles lies.

A quasi compact and quasi-separated scheme has its derived category of sheaves of modules with quasi-coherent cohomology generated by perfect complexes. This is actually a theorem of Bondal and Van de Berg Generators and representability of functors in commutative and noncommutative geometry. However, there are non-separated schemes that, despite the previous theorem, do not posses the resolution property, see the paper by Totaro The resolution property for schemes and stacks for a discussion of the subtleties of the issue.

A quasi compact and quasi-separated scheme has its derived category of sheaves of modules with quasi-coherent cohomology generated by perfect complexes. This is actually a theorem of Bondal and Van de Berg Generators and representability of functors in commutative and noncommutative geometry. However, there are non-separated schemes that, despite the previous theorem, do not posses the resolution property, see the paper by Totaro The resolution property for schemes and stacks for a discussion of the subtleties of the issue.

The proof by Bondal-Van den Bergh uses Thomason-Neeman localization: a perfect complex overt a dense open subset extends. This does not hold for vector bundles in general. This is where the difference between compact generation and generation by vector bundles lies.

A quasi compact and quasi-separated scheme has its derived category of sheaves of modules with quasi-coherent cosmology generated by perfect complexes. This is actually a theorem of Bondal and Van de Berg Generators and representability of functors in commutative and noncommutative geometry. However, there are non-separated schemes that, despite the previous theorem, do not posses the resolution property, see the paper by Totaro The resolution property for schemes and stacks for a discussion of the subtleties of the issue.

A quasi compact and quasi-separated scheme has its derived category of sheaves of modules with quasi-coherent cohomology generated by perfect complexes. This is actually a theorem of Bondal and Van de Berg Generators and representability of functors in commutative and noncommutative geometry. However, there are non-separated schemes that, despite the previous theorem, do not posses the resolution property, see the paper by Totaro The resolution property for schemes and stacks for a discussion of the subtleties of the issue.