= Rank bound for locally free ideals on reduced schemes
A finite <locally free sheaf> of ideals on a <reduced scheme> has rank at most one at every point. On a nonempty <affine open subscheme> where its rank is $r$, localize its inclusion into the structure sheaf at a <minimal prime ideal>. The resulting <local ring> is a field $K$, giving an injection $K^r\hookrightarrow K$, hence $r\le1$. This works without a Noetherian assumption. The empty scheme is a vacuous exception to claims phrased as nonexistence of a sheaf of a prescribed rank.
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