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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.

Brenin
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1 Answers1

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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.
All the other adjectives (rank-one, coherent, smooth, projective, irreducible,...) are irrelevant.

Also, you shouldn't believe that ideal sheaves must be of rank one or quasi- coherent :
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 .

Georges Elencwajg
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  • 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
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    Dear @Brenin: yes, my example is pathological and non-geometric: I should have emphasized this in my answer. In the geometric context of varieties or schemes over a field, ideal sheaves tend however to be coherent and rank one, unless one deliberately considers nasty examples. Nice and fairly typical rank one examples are given by the ideal sheaves corresponding to closed subvarieties. – Georges Elencwajg Oct 15 '13 at 18:29