Dualizable quasicoherent sheaves are finite locally free

On an arbitrary scheme, a quasicoherent sheaf has a left or right dual in the symmetric monoidal category of quasicoherent sheaves if and only if it is finite locally free. No affine, quasi-compactness, separation or noetherian hypothesis is needed.

Pullback preserves dualizability, so a dualizable sheaf restricts to a dualizable sheaf on each affine chart. The affine criterion identifies its module of sections as finite projective; finite local freeness then descends along the affine open cover. In the converse direction, the internal-Hom dual of a finite locally free sheaf supplies its categorical dual.

References

• The Stacks Project, Sheaves of Modules, finite locally free modules and duality.

theorem TauCeti.AlgebraicGeometry.QuasicoherentSheaf.nonempty_hasLeftDual_iff_isFiniteLocallyFree {X : AlgebraicGeometry.Scheme} (E : QuasicoherentSheaf X) : Nonempty (CategoryTheory.HasLeftDual E) ↔ AlgebraicGeometry.Scheme.Modules.isFiniteLocallyFree X E.obj

A quasicoherent sheaf on any scheme has a left dual in QuasicoherentSheaf X if and only if it is finite locally free.

theorem TauCeti.AlgebraicGeometry.QuasicoherentSheaf.nonempty_hasRightDual_iff_isFiniteLocallyFree {X : AlgebraicGeometry.Scheme} (E : QuasicoherentSheaf X) : Nonempty (CategoryTheory.HasRightDual E) ↔ AlgebraicGeometry.Scheme.Modules.isFiniteLocallyFree X E.obj

A quasicoherent sheaf on any scheme has a right dual in QuasicoherentSheaf X if and only if it is finite locally free.