Suppose $X$ is a $n$ dimensional proper smooth variety, is the dualizing sheaf of $X$ the top wedge of sheaf of differentials: $\omega\_X^0=\wedge^n\Omega^1\_X$? If not what is it? 


(By Chow lemma, we have a dominating morphism from a smooth projective variety $X'\to X$. Can we construct the dualizing sheaf from this?)



The dualizing sheaf for a proper smooth variety
As a complement to abx's comment, and since it is somehow difficult to locate the exact statement in Hartshorne's *Residues and Duality*, let me point out the precise result (or better a relative version of it), that can be found for instance in [1](https://eudml.org/doc/89450), Proposition 22 p. 55.



> 
> **Theorem.** Let $f \colon X \to S$ be a flat Cohen-Macauley morphism of schemes (i.e., all fibres of $f$ are Cohen-Macauley schemes), and let $V \subset X$ be the smooth locus of $f$. Then there is a canonical isomorphism $$\omega\_f|\_V = \det \Omega^1\_{V|S},$$
>  where $\omega\_f$ denotes the relative dualizing sheaf with respect to $f$. 
> 
> 
> 


In particular, when $S = \textrm{Spec}(k)$ and $X$ is any smooth scheme, we deduce the desired isomorphism $$\omega\_X = \det \Omega^1\_{X} = \bigwedge^{\dim X} \Omega\_X^1.$$


**References.**


[1](https://eudml.org/doc/89450) Steven L. Kleiman: *[Relative duality for quasi-coherent sheaves](https://eudml.org/doc/89450)*, Compositio Mathematica **41** (1980), 39-60.


