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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.

k.j.
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  • See https://math.stackexchange.com/questions/679768/projective-noether-normalization. I also wrote the proof out in the affine case on my blog here: https://asving.com/2017/09/11/noether-normalization-spreading-out-and-the-nullstellensatz/ , it is theorem 2. I only do the affine case but the same proof will prove the projective case too. – Asvin Nov 29 '17 at 11:37
  • @Asvin Thank you so much, but I couldn't.... Please show me more detail. Especially I don't know why the morphism is separable. – k.j. Nov 29 '17 at 14:49
  • Hmm, I missed the separable part but if you want the map to be separable, then I think you should insist on X being geometrically reduced. Because if there is map like you want, then it implies X is geometrically reduced... – Asvin Nov 29 '17 at 22:12
  • @Asvin So, is this pdf wrong? But it is no problem to add the condition that X is geometrically reduced, since I want to prove the Riemann-Roch for the smooth curve. – k.j. Nov 29 '17 at 22:18
  • @Asvin Oh, now $k$ is perfect and $X$ is integral, so this is geometrically reduced. – k.j. Nov 29 '17 at 22:23
  • Yes, you are right. I missed the perfect condition too... I think I know how to do it then but I am on mobile, I will try to write something in a couple of hours... Basically the idea is that if the map is not separable, look at the field extension at the generic point, in this field extension, you can find new transcendental variables so that it is a separable extension and the way you find these new variables lets you construct the map to P^n from X too. This is probably too vague... – Asvin Nov 29 '17 at 22:30
  • @Asvin Thank you, I keep trying this lemma and look forward. – k.j. Nov 29 '17 at 22:38
  • I found a reference that proves what you want : http://therisingsea.org/notes/MoreNoetherNormalisation.pdf – Asvin Nov 29 '17 at 23:30

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