Lemma 53.6.1. Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \mathcal{O}_ X) = k$. Then $X$ is connected, Cohen-Macaulay, and equidimensional of dimension $1$.
Proof. Since $\Gamma (X, \mathcal{O}_ X) = k$ has no nontrivial idempotents, we see that $X$ is connected. This already shows that $X$ is equidimensional of dimension $1$ (any irreducible component of dimension $0$ would be a connected component). Let $\mathcal{I} \subset \mathcal{O}_ X$ be the maximal coherent submodule supported in closed points. Then $\mathcal{I}$ exists (Divisors, Lemma 31.4.6) and is globally generated (Varieties, Lemma 33.33.3). Since $1 \in \Gamma (X, \mathcal{O}_ X)$ is not a section of $\mathcal{I}$ we conclude that $\mathcal{I} = 0$. Thus $X$ does not have embedded points (Divisors, Lemma 31.4.6). Thus $X$ has $(S_1)$ by Divisors, Lemma 31.4.3. Hence $X$ is Cohen-Macaulay. $\square$
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