Dualizing sheaf
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 ( X , ω X ) → k {\displaystyle t_{X}:\operatorname {H} ^{n}(X,\omega _{X})\to k} that induces a natural isomorphism of vector spaces Hom X ( F , ω X ) ≃ H n ( X , F ) ∗ , φ ↦ t X ∘ φ {\displaystyle \operatorname {Hom} _{X}(F,\omega _{X})\simeq \operatorname {H} ^{n}(X,F)^{*},\,\varphi \mapsto t_{X}\circ \varphi } for each coherent sheaf F on X (the superscript * refers to a dual vector space). The linear functional t X {\displaystyle t_{X}} is called a trace morphism.