Exceptional subscheme of a blowup

Let $X$ be a scheme, let $\mathcal{I}\subseteq {\mathcal{O}}_X$ be a quasi-coherent ideal sheaf (Quasi-coherent ideal sheaves) with zero scheme $Z=V(\mathcal{I})\hookrightarrow X$, the closed subscheme cut out by $\mathcal{I}$ (Closed immersions of schemes), and let $\pi \text{\hspace{0.17em}⁣}:{\mathrm{Bl}}_{\mathcal{I}}X\to X$ be the blowup of Blowup of a scheme along an ideal sheaf. The exceptional subscheme of the blowup is the scheme-theoretic inverse image

$E:={\pi }^{-1}(Z)=Z{\times }_X{\mathrm{Bl}}_{\mathcal{I}}X$

of Scheme-theoretic inverse images of subschemes (Base change of objects, morphisms and properties); it is a closed subscheme $E\hookrightarrow {\mathrm{Bl}}_{\mathcal{I}}X$, its ideal sheaf is the inverse image ideal $\mathcal{I}{\mathcal{O}}_{{\mathrm{Bl}}_{\mathcal{I}}X}:=\mathrm{Im}({\pi }^{\ast }\mathcal{I}\to {\mathcal{O}}_{{\mathrm{Bl}}_{\mathcal{I}}X})$, and the structural morphism of the blowup restricts to a morphism $E\to Z$. Set-theoretically, $E$ is the preimage ${\pi }^{-1}(Z)$ of the underlying set of $Z$.

For an ideal not of finite type, the notation here extends Blowup of a scheme along an ideal sheaf by using ${\mathrm{Proj}}_X({⨁}_{n\ge 0}{\mathcal{I}}^n)$ directly. Its graded algebra is quasi-coherent by the affine ideal-power calculation of Rees algebra sheaf of a finite type ideal, and no finite generation is required by Relative Proj of a graded quasi-coherent algebra.

• The exceptional subscheme is defined for every quasi-coherent ideal sheaf $\mathcal{I}$ on $X$; no smoothness of $X$, regularity of $Z$, or invertibility of $\mathcal{I}$ is assumed.
• The construction inherits the Axiom of Choice from the blowup (Blowup of a scheme along an ideal sheaf), and no further data is chosen.
• Nothing is asserted here about the components of $E$, its codimension in the blowup, or the invertibility of its ideal sheaf; those are supplied by later items of this page.