subscheme of blow up

Question

Let $X$ be a noetherian scheme, $Y$ a closed subscheme defined by ideal sheaf $\mathcal{I}$ and $\pi : \widetilde{X} \rightarrow X$ be the blowing up of $X$ along $Y$. Let $Y'$ be the subscheme of $\widetilde{X}$ defined by the ideal sheaf $\mathcal{I}' = \pi^{-1}\mathcal{I} \cdot \mathcal{O}_{\widetilde{X}}$

In Hartshorne book Theorem (8.24), since $\widetilde{X}= \operatorname{Proj} \bigoplus_{d\geq 0} \mathcal{I}^d$, $Y'\cong \operatorname{Proj} \bigoplus_{d\geq 0}(\mathcal{I}^d\otimes \mathcal{O}_X/\mathcal{I})$. But I don't understand this fact.....

Answer 1

Here are two simple facts:

• The Proj construction is stable under base change. By this I mean that for every graded quasi-coherent algebra $A$ on $X$ and every morphism $f : Y \to X$, we have $\mathrm{Proj}(A) \times_X Y = \mathrm{Proj}(f^* A)$. This follows from the explicit construction, or by comparing the universal properties of both schemes.

• If $p : P \to X$ is a morphism, $I \subseteq \mathcal{O}_X$ is a quasi-coherent ideal and $I' := p^{-1} I \cdot \mathcal{O}_P$, then $P \times_X V(I) \cong V(I')$. Locally, this is just the well-known isomorphism $A \otimes_R R/I = A/IA$ for an $R$-algebra $A$ and an ideal $I \subseteq A$.

Now, if $\pi : \tilde{X} \to X$ is the blow up along $V(I)$, applying the first fact to $V(I) \hookrightarrow X$ and the second fact to $\pi$, we see that $V(I') = \mathrm{Proj}(\oplus_{d \geq 0} I^d \otimes \mathcal{O}_X/I)$.