In algebraic geometry, the dualizing sheaf on a proper scheme X of dimension n over a field k is a coherent sheaf ω X {\displaystyle \omega _{X}} together with a linear functional t X : H n ⁡ → k {\displaystyle t_{X}:\operatorname {H} ^{n}\to k} that induces a natural isomorphism of vector spaces Hom X ⁡ ≃ H n ⁡ ∗, φ ↦ t X ∘ φ {\displaystyle \operatorname {Hom} _{X}\simeq \operatorname {H} ^{n}^{*},\,\varphi \mapsto t_{X}\circ \varphi } for each coherent sheaf F on X. Wikipedia