The ideal sheaf of a closed subscheme of the projective $\mathbb{C}$-scheme.

Question

Consider the closed subscheme $\text{Spec}(\mathbb{C})\sqcup\text{Spec}(\mathbb{C})$ of $\mathbb{P}_{\mathbb{C}}^{1}$. Let $i: \text{Spec}(\mathbb{C})\sqcup\text{Spec}(\mathbb{C}) \rightarrow \mathbb{P}_{\mathbb{C}}^{1}$ be the corresponding closed immersion. For simplicity we write $X=\mathbb{P}^{1}_{\mathbb{C}}$. I want to show that $i_{*}\mathcal{O}_{\text{Spec}(\mathbb{C})\sqcup\text{Spec}(\mathbb{C})}$ is isomorphic to $\widetilde{M}$, where $M = S/I$ with $S = \mathbb{C}[X_{0},X_{1}]$ and $I = (X_{0}X_{1})$. And that $\widetilde{I}$ is the ideal sheaf of $\text{Spec}(\mathbb{C})\sqcup\text{Spec}(\mathbb{C})$

Answer 1

A hint: using the fact that $I$ is the homogeneous ideal defining your closed subscheme, try to prove directly that $\tilde I$ is the corresponding ideal sheaf. You can do this by looking at distinguished open affines and comparing the definition of the ideal sheaf to the definition of the quasicoherent sheaf $\tilde I$ - the sections and the restriction maps can be explicitly described nicely in this example.

(I don't have enough reputation to post this as a comment unfortunately)