Strict transforms of closed subschemes are blowups of the subscheme

By [F2] the inverse image ideal $\mathcal{J}=\mathcal{I}{\mathcal{O}}_W$ is quasi-coherent of finite type with zero scheme $W{\times }_{X}Z$, so the blowup ${\mathrm{Bl}}_{\mathcal{J}}W$ and its structural morphism $\sigma \text{\hspace{0.17em}⁣}:{\mathrm{Bl}}_{\mathcal{J}}W\to W$ are defined; composing with $W\to X$ exhibits it as an $X$-scheme. By [F3] the inverse image of $Z$ on ${\mathrm{Bl}}_{\mathcal{J}}W$ is cut by the invertible ideal $\mathcal{J}{\mathcal{O}}_{{\mathrm{Bl}}_{\mathcal{J}}W}$, hence is an effective Cartier divisor, so the universal property supplies a unique $X$-morphism $\psi \text{\hspace{0.17em}⁣}:{\mathrm{Bl}}_{\mathcal{J}}W\to {\mathrm{Bl}}_{\mathcal{I}}X$. Together with $\sigma$ it defines a morphism $u=(\sigma ,\psi )\text{\hspace{0.17em}⁣}:{\mathrm{Bl}}_{\mathcal{J}}W\to W{\times }_X{\mathrm{Bl}}_{\mathcal{I}}X$ over $X$.

Chart computation. Let $\mathrm{Spec}A\subseteq X$ be affine with $I=(f_0,\dots ,f_r)$ and $W=\mathrm{Spec}(A\mathrm{/}K)$. Write $B_i=A[I\mathrm{/}{f}_i]$ and $C_i=(A\mathrm{/}K)[J\mathrm{/}{\bar{f}}_i]$, so that $\mathrm{Spec}{C}_i$ is the chart of ${\mathrm{Bl}}_{\mathcal{J}}W$ over $\mathrm{Spec}(A\mathrm{/}K)$. The morphism $u$ on this chart is the $A$-algebra homomorphism ${\phi }_i\text{\hspace{0.17em}⁣}:B_i\to C_i$ with ${\phi }_i(a)=\bar{a}$ for $a\in A$ and ${\phi }_i(f_j\mathrm{/}{f}_i)={\bar{f}}_j\mathrm{/}{\bar{f}}_i$. It is surjective because $C_i$ is generated over $A\mathrm{/}K$ by the elements ${\bar{f}}_j\mathrm{/}{\bar{f}}_i$. Its kernel is the saturation $(K{B}_i:f_i^{\mathrm{\infty }})$: indeed $b\in \ker {\phi }_i$ if and only if the image of $b$ in $C_i[1\mathrm{/}{\bar{f}}_i]$ vanishes, and by [F4] applied to $A\mathrm{/}K$ and $J$ this localized ring is $(B_i\mathrm{/}K{B}_i)[1\mathrm{/}{f}_i]$, the localization of $B_i\mathrm{/}K{B}_i$ at $f_i$, so $b$ lies in the kernel precisely when $f_i^{n}b\in K{B}_i$ for some $n$, which is the saturation.

By step 2.1 the restriction of $u$ to the chart $\mathrm{Spec}{C}_i$ is the closed immersion $\mathrm{Spec}{C}_i\hookrightarrow \mathrm{Spec}{B}_i$ cut out by the saturation $(K{B}_i:f_i^{\mathrm{\infty }})$, whose image is exactly the piece of $W^{\mathrm{\prime }}$ over $\mathrm{Spec}A$ described in [F1]; this is the full preimage of chart $i$: on a source chart $j$, membership in target chart $i$ means the pulled-back ratio ${\bar{f}}_i\mathrm{/}{\bar{f}}_j$ is a unit, precisely the overlap with source chart $i$. Equivalently, if the image lies in target chart $i$, the pullback of its center ideal is generated by ${\bar{f}}_i$; comparison with a regular generator ${\bar{f}}_j$ on a source chart forces their ratio to be a unit, as in the universal-property proof. Thus $\mathrm{Spec}{C}_i$ is the full preimage. Therefore these chart descriptions agree on overlaps and cover the target, so by [F5] the morphism $u$ is a closed immersion and its image is exactly $W^{\mathrm{\prime }}$. Hence $u$ identifies $W^{\mathrm{\prime }}$ with ${\mathrm{Bl}}_{\mathcal{J}}W$ over $W$, and in particular $W^{\mathrm{\prime }}$ is the blowup of $W$ along $\mathcal{J}=\mathcal{I}{\mathcal{O}}_W$, equivalently along $W\cap Z$.

Canonicity. The morphism $\psi$ is the unique $X$-morphism from ${\mathrm{Bl}}_{\mathcal{J}}W$ to ${\mathrm{Bl}}_{\mathcal{I}}X$ provided by the universal property in [F3], and $\sigma$ is the structural morphism of the blowup, so $u$ is determined by the data of the two blowups; conversely the inverse $W^{\mathrm{\prime }}\to {\mathrm{Bl}}_{\mathcal{J}}W$ is obtained by gluing the inverse chart isomorphisms $B_i\mathrm{/}(K{B}_i:f_i^{\mathrm{\infty }})\cong C_i$ from steps 2.1–3.1, determined by the same data, and any two isomorphisms with these properties agree by [F6]. Thus the identification of $W^{\mathrm{\prime }}$ with ${\mathrm{Bl}}_{\mathcal{J}}W$ is canonical.

Union statement. Suppose $W=W_1\cup W_2$ is the scheme-theoretic union of two closed subschemes, so on an affine chart their ideals satisfy $K=K_1\cap K_2$. The saturation of $K$ with respect to $f_i$ is the preimage of the ideal $K{B}_i[1\mathrm{/}{f}_i]$ under $B_i\to B_i[1\mathrm{/}{f}_i]$ by [F4], and $B_i[1\mathrm{/}{f}_i]=A_{f_i}$. Flat localization gives $(K_1\cap K_2)A_{f_i}=K_1{A}_{f_i}\cap K_2{A}_{f_i}$; taking preimages under $B_i\to A_{f_i}$ commutes with finite intersections; hence $(K{B}_i:f_i^{\mathrm{\infty }})=(K_1{B}_i:f_i^{\mathrm{\infty }})\cap (K_2{B}_i:f_i^{\mathrm{\infty }})$ on every chart. By step 2.1 the right-hand side cuts out the union of the chart pieces of $W_1^{\mathrm{\prime }}$ and $W_2^{\mathrm{\prime }}$, and these chartwise identifications glue, so $W^{\mathrm{\prime }}=W_1^{\mathrm{\prime }}\cup W_2^{\mathrm{\prime }}$ as closed subschemes of $W{\times }_X{\mathrm{Bl}}_{\mathcal{I}}X$. Induction gives the result for every finite scheme-theoretic union. In particular a finite union of components is transformed componentwise, and $W^{\mathrm{\prime }}$ is the union of the strict transforms of its parts; this union may be empty or have a single component.