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

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 $$\text{Spec}(\mathbb{C})\sqcup\text{Spec}(\mathbb{C})$$

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.
• When you restrict to $U_0\cap U_1 = {\rm Spec} (\mathbb C[X_0,X_1]_{X_0 X_1})_0$, you should end up with $\mathcal I(U_0\cap U_1) = (X_{10})_{X_{10}}$ and $\tilde I(U_0\cap U_1) = (I_{X_0 X_1})_0$. Now use the isomorphism $(\mathbb C[X_0,X_1]_{X_0 X_1})_0 \cong \mathbb C[X_{10}]_{X_{10}}$. – Alex K Apr 27 at 13:18