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Let $X$ be a noetherian scheme and $\mathcal I$ a sheaf of ideals of $\mathcal O_X$. Is $\mathcal I$ a quasi-coherent sheaf on $X$?

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If I remember my algebraic geometry correctly, this should not hold in general, however, it holds precisely if $\mathcal{I}$ corresponds to a closed embedding of another scheme. Vakil has in his notes beautiful examples and exercises about precisely this subtlety, which I can only recommend: http://math.stanford.edu/~vakil/216blog/FOAGjun1113public.pdf

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