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Exercise 4.1.3 of Qing Liu's Algebraic Geometry and Arithmetic Curves asks of us to prove, given a normal Noetherian local scheme $X$ of dimension $2$ with closed point $s$, that $X \backslash \{ s \}$ is a non-affine Dedekind scheme.

However, a Dedekind scheme has dimension $1$ and Liu defines normal schemes to be irreducible, so by taking the reduced induced subscheme structure on $X$, which is integral, we see that the open subset $X \backslash \{ s \}$ must also have dimension $2$ and so cannot be Dedekind. Surely even if I relax the definition of normal scheme to include reducible schemes, the result wouldn't hold in the irreducible case anyway? Have I missed something?

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  • $\begingroup$ If local scheme means spectrum of a local ring, then surely removing the closed point drops the dimension by 1. Maybe work out what happens with $k[x,y]_{(x,y)}$. $\endgroup$ – Samir Canning Jul 17 '18 at 15:52
  • $\begingroup$ You are of course correct, I have made the error of assuming the function field argument works for schemes which are not algebraic varieties over a field. Indeed, just be looking at the spectrum of any non-trivial DVR we see that the generic point is open but has dimension $0$. $\endgroup$ – Rob Rockwood Jul 18 '18 at 8:13

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