Let $k$ be a field, $X=\mathbb{P}_k^n$ be projective space, and let $Y\subset X$ be some irreducible codimension one sub variety (i.e. a hypersurface). Now, I know that we must be able to express $Y$ in the form $\operatorname{Proj}\, (k[x_0,\ldots,x_n]/\mathfrak{a})$ for some $\mathfrak{a}$, but must it be the case that $\mathfrak{a}$ is principal? I know that if we replace Proj with Spec, this is true.
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$\begingroup$Choose a homogeneous element of $\mathfrak a$ and take one of its irreducible factors, say $f$. Using the lexicographic order on monomials, it is easy to prove that $f$ should be homogeneous. We can choose $f$ vanishing on $Y$, then we have $Y\subset V(f)$, an inclusion of irreducible varieties of the same dimension, therefore $Y=V(f)$.
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