I am reading a lot about curves at the moment and I am a little confused: Let $X= Spec K[X]$ denote a smooth affine algebraic curve. Then, according to some sources, the ring $K[X]$ is a Dedekind domain. But this would mean $X$ to constist of only two points, the generic point and a special point...is this correct?? How can the curve constist of only two points??
Also, considering a subscheme $Y \hookrightarrow X$, is the coordinate ring of $Y$ again a Dedekind domain (is $Y$ a curve?). The only way $Y$ could be a nontrivial subscheme would be for $K[Y]$ to be a field, consisting of only one prime ideal $(0)$.
Am I mistaken at some point here?
I'm a little bit confused!
But I expose an easy example: let $X=\mathbb{A}^1_{\mathbb{K}}$ be the affine line over an integrally closed field $\mathbb{K}$, then $X$ as affine scheme is $Spec\mathbb{K}[t]=\{(t-a)\,\text{prime ideal of}\,\mathbb{K}[t]:a\in\mathbb{K}\}\cup\{(0)\}$; by hypothesis, $\mathbb{K}$ has infinitely many elements, and $\mathbb{K}[x]$ is a Dedekind domain!, indeed: $\mathbb{K}[x]$ is integrally closed and Noetherian ring, and its Krull dimension is $1$.
If you take a closed point $y$ of $X$, you find that $\{y\}=Spec\mathbb{K}[t]_{\displaystyle/\mathfrak{m}_y}\cong Spec\mathbb{K}=\{(0)\}$ the zero ideal of $\mathbb{K}$.