# Rank one sheaves and ideal sheaves

For a coherent sheaf $\mathcal F$ on a smooth irreducible projective variety $X/k$, it makes sense to define the rank $\textrm{rk }\mathcal F$ as the rank of the vector bundle $\mathcal F|_U$, where $U$ is the open subset of $X$ where $\mathcal F$ is locally free.

Ideal sheaves $\mathscr I\subset\mathcal O_X$ are coherent of rank one.

Question. Is there a known criterion saying when a coherent subsheaf $\mathcal F\subset \mathcal O_X$ of rank one is an ideal sheaf?

Thanks for any suggestion, or reference.

Any subsheaf of $\mathcal O_X$-modules $\mathcal F\subset \mathcal O_X$ on a scheme (or even on a ringed space) is an ideal sheaf.
On the spectrum $X=\text {Spec} R$ of a discrete valuation ring $R$, consider the ideal sheaf $\mathcal I$ with global sections $\Gamma(X,\mathcal I)=R$ and whose sections over the (open!) generic point are given by $\Gamma(\{\eta\},\mathcal I)=0$.
The sheaf $\mathcal I$ is an ideal sheaf which is not quasi-coherent and which is of rank zero .
• Dear @GeorgesElencwajg, your answers are always illuminating. May I just ask you "how much" your example is pathological? For instance: can we have rank $0$ ideal sheaves $I\subset O_X$ for $X$ a projective variety over a field $k=\overline k$? let us say one looks at ideals of curves in $X$, or curves and points...? For sure, ideals of zero-dimensional subschemes are rank-one! – Brenin Oct 15 '13 at 15:44