# Is $\mathcal I$ a quasi-coherent sheaf on $X$?

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 $$X$$?

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