Blowups restrict to open subschemes of the base

The pullback along $j$ of the Rees algebra is the Rees algebra of the restricted ideal: $j^{\ast }\mathcal{R}(\mathcal{I})\cong \mathcal{R}(\mathcal{I}{\mathrm{\mid }}_U)$ as graded ${\mathcal{O}}_U$-algebras. Indeed, restriction to the open subscheme $U$ is exact and commutes with tensor products, so $j^{\ast }({\mathcal{I}}^n)\cong (j^{\ast }\mathcal{I}{)}^n=(\mathcal{I}{\mathrm{\mid }}_U{)}^n$ for every $n\ge 0$, and these identifications are compatible with the multiplications inherited from ${\mathcal{O}}_X$ and ${\mathcal{O}}_U$; the graded pieces of both sides are quasi-coherent by [F3], and $\mathcal{I}{\mathrm{\mid }}_U$ is again quasi-coherent (of finite type when $\mathcal{I}$ is).

Applying [F1] to the morphism $j\text{\hspace{0.17em}⁣}:U\to X$ and the graded algebra $\mathcal{A}=\mathcal{R}(\mathcal{I})$ gives a canonical isomorphism of $U$-schemes ${\mathrm{Bl}}_{\mathcal{I}}X{\times }_{X}U={\mathrm{Proj}}_X\mathcal{R}(\mathcal{I}){\times }_{X}U\cong {\mathrm{Proj}}_U(j^{\ast }\mathcal{R}(\mathcal{I}))\cong {\mathrm{Proj}}_U\mathcal{R}(\mathcal{I}{\mathrm{\mid }}_U)={\mathrm{Bl}}_{\mathcal{I}{\mathrm{\mid }}_U}U$, where the last equality is the definition of the blowup of $U$ along $\mathcal{I}{\mathrm{\mid }}_U$; the isomorphism is compatible with the relative twists.

The first projection ${\mathrm{Bl}}_{\mathcal{I}}X{\times }_{X}U\to {\mathrm{Bl}}_{\mathcal{I}}X$ is the base change of the open immersion $j$ along $\pi$, hence an open immersion by [F4], and its underlying image is the open subset ${\pi }^{-1}(U)$. Identifying the fibre product with this open subscheme via that open immersion turns the isomorphism of step 2.1 into an isomorphism ${\pi }^{-1}(U)\to {\mathrm{Bl}}_{\mathcal{I}{\mathrm{\mid }}_U}U$ over $U$.

The isomorphisms are compatible with inclusions of opens: for open subschemes $U^{\mathrm{\prime }}\subseteq U\subseteq X$ one has $({\mathrm{Bl}}_{\mathcal{I}}X{\times }_{X}U){\times }_U{U}^{\mathrm{\prime }}={\mathrm{Bl}}_{\mathcal{I}}X{\times }_X{U}^{\mathrm{\prime }}$ by [F5], and the isomorphism of [F1] is natural in the base morphism, so the identifications for $U$ and for $U^{\mathrm{\prime }}$ restrict to one another; the same naturality makes the passage to ${\pi }^{-1}(U)$ of step 3.1 compatible with the inclusion ${\pi }^{-1}(U^{\mathrm{\prime }})\subseteq {\pi }^{-1}(U)$.