Ideal sheaf is aquasi-coherent sheaf of ideals.

Question

Let $X$ be a scheme, For any closed subscheme $Y$ of $X$, the corresponding ideal $I_Y$ given by the kernel of the morphism $i^{\#}:\mathcal{O}_x\rightarrow i_{*}\mathcal{O}_y$ is quasi-coherent sheaf of ideals on X.

I´m reading something about Tools for birrational geometry, and this is a proposition without proof, and I want some refence (not hartshorne) where I can read the proof of this.

Thanks.

Answer 1

Just to give this an answer: there are proofs at Stacks project tag 01QO and in II, 5.9 of Hartshorne. The proof in Hartshorne is not so bad. I think most proofs will use two ingredients: $\textsf{QCoh}_X$ is an abelian subcategory of $\textsf{Mod}_X$; and if you have a morphism $f\colon X \to Y$ then the functor $f_*$ doesn't always take $\textsf{QCoh}_X$ to $\textsf{QCoh}_Y$, but it does if the morphism is affine or more generally if it is quasicompact.