Lemma 33.34.4. Let $k$ be a field. Let $Z \subset \mathbf{P}^ n_ k$ be a closed subscheme which has no embedded points such that every irreducible component of $Z$ has dimension $n - 1$. Then the ideal $I(Z) \subset k[T_0, \ldots , T_ n]$ corresponding to $Z$ is principal.

Proof. This is a special case of Divisors, Lemma 31.31.3. $\square$

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