Now I read http://math.stanford.edu/~vakil/216blog/Baker-Csirik-serre-duality.pdf to prove that the dualizing sheaf is isomorphic to the canonical sheaf, for smooth projective scheme over a perfect field. And to show this, I have to show the lemma2 of this pdf. This pdf says the proof is in Shafarevich's Basic Algebraic Geometry chapter5 theorem9, but I can't find such theorem.
So please show the lemma, or give some reference.
This is the lemma:
For a perfect field $k$ and an integral projective $k$-scheme $X$ of dimension $n$, there exists a finite separable morphism $ X \to \mathbb{P}^n_k. $
And please give some references of the dualizing sheaf and the canonical sheaf over a non-algebraically closed field. I want to show the Riemann-Roch theorem for smooth projective schemes over a perfect field, but Hartshorne proves only the case of algebraically closed fields, and Liu does not show the existence of dualizing sheaf over finite fields.