Characterisation of closed subschemes of projective spaces

Question

I'm trying to understand a step in the proof of corollary 5.16 in Hartshorne. The statement

Let $ A $ be a ring. if $ Y$ is a closed subscheme of $\Bbb{P}_{A}^{r} $ then there is a homogeneous ideal $ I\subset S=A[x_{0},...x_{r}]$ such that $ Y $ is the closed subscheme determined by $ I $.

The proof takes the quasi coherent ideal sheaf $ \mathcal{I}$ corresponding to $Y $ and shows that $\Gamma_*(\mathcal {I})\subset\Gamma_*(\mathcal{O}_{X}) = S$. Then sets $ I=\Gamma_*(\mathcal {I})$ and uses the isomorphism $\tilde{I}\cong\mathcal{I}$ proved earlier to conclude that $ I $ induces the same closed subscheme.

The parts that are unclear to me:

1) How do we see that $\tilde{I}$ is an ideal sheaf?

2) I feel that we need to show that $\tilde{I}$ and $\mathcal{I}$ are the same ideal sheaf. It isn't sufficient to show that they are isomorphic. The correspondence we have is between closed subschemes and quasi coherent ideal sheaves. It isn't between closed subschemes and isomorphism classes of quasi coherent ideal sheaves. What am I missing?

Answer 1

The important thing here is that we have:

1. A canonical way to identify $\tilde{I}$ with $\mathcal I$; and
2. A canonical way to embed $\tilde{I}$ into $O_X=\tilde{S}$. (This answers your first question)

The proof shows that these two maps are actually the same.