However if X is proper and finite type over any field there does exist a dualizing sheaf, and it agrees with the top nonzero homology sheaf of the dualizing complex.

Addendum

I mean by dualizing sheaf, a sheaf that represents the functor $\mathcal{F} \mapsto H^n(X,\mathcal{F})^{\vee}$, i.e. such that there is an ismorphism

$$ \mathrm{Hom}_X(\mathcal{F},\omega_X) \cong H^n(X,\mathcal{F})^{\vee}. $$

For me, canonical is related to local duality, in this sense $\omega_{X,x}$ is a canonical module for the ring $\mathcal{O}_{X,x}$.

Of course you need Cohen Macaulay to have

$$ \mathrm{Ext}^i_X(\mathcal{F},\omega_X) \cong H^{n-i}(X,\mathcal{F})^{\vee}. $$ for every $i$; this would make $\omega_X$ a dualizing complex. This story is explained in a somewhat elementary way (without derived categories) in Kleiman's "Relative duality for quasi-coherent sheaves" in Compositio Mathematica. In the case that $k$ is a perfect field there is characterization of $\omega_X$ in terms of differentials and traces, see Lipman's Asterisque 117, aka "Lipman's blue book". In this book a nice treatment of residues is given in this setting.

However if $X$ is proper and finite type over any field there does exist a dualizing sheaf, and it agrees with the top nonzero homology sheaf of the dualizing complex.

Addendum

I mean by dualizing sheaf, a sheaf that represents the functor $\mathcal{F} \mapsto H^n(X,\mathcal{F})^{\vee}$, i.e. such that there is an ismorphism

$$ \mathrm{Hom}_X(\mathcal{F},\omega_X) \cong H^n(X,\mathcal{F})^{\vee}. $$

For me, canonical is related to local duality, in this sense $\omega_{X,x}$ is a canonical module for the ring $\mathcal{O}_{X,x}$.

Of course you need Cohen Macaulay to have

$$ \mathrm{Ext}^i_X(\mathcal{F},\omega_X) \cong H^{n-i}(X,\mathcal{F})^{\vee}. $$ for every $i$; this would make $\omega_X$ a dualizing complex. This story is explained in a somewhat elementary way (without derived categories) in Kleiman's "Relative duality for quasi-coherent sheaves" in Compositio Mathematica. In the case that $k$ is a perfect field there is characterization of $\omega_X$ in terms of differentials and traces, see Lipman's Asterisque 117, aka "Lipman's blue book". In this book a nice treatment of residues is given in this setting.

However if X is proper and finite type over a perfect field there does exist a dualizing sheaf, and it agrees with the top nonzero homology sheaf of the dualizing complex.

However if X is proper and finite type over any field there does exist a dualizing sheaf, and it agrees with the top nonzero homology sheaf of the dualizing complex.

Addendum

I mean by dualizing sheaf, a sheaf that represents the functor $\mathcal{F} \mapsto H^n(X,\mathcal{F})^{\vee}$, i.e. such that there is an ismorphism

$$ \mathrm{Hom}_X(\mathcal{F},\omega_X) \cong H^n(X,\mathcal{F})^{\vee}. $$

For me, canonical is related to local duality, in this sense $\omega_{X,x}$ is a canonical module for the ring $\mathcal{O}_{X,x}$.

Of course you need Cohen Macaulay to have

$$ \mathrm{Ext}^i_X(\mathcal{F},\omega_X) \cong H^{n-i}(X,\mathcal{F})^{\vee}. $$ for every $i$; this would make $\omega_X$ a dualizing complex. This story is explained in a somewhat elementary way (without derived categories) in Kleiman's "Relative duality for quasi-coherent sheaves" in Compositio Mathematica. In the case that $k$ is a perfect field there is characterization of $\omega_X$ in terms of differentials and traces, see Lipman's Asterisque 117, aka "Lipman's blue book". In this book a nice treatment of residues is given in this setting.