Universal property of the blowup

Given: The Axiom of Choice, a quasi-coherent ideal sheaf $\mathcal{I}$ of finite type on $X$ with zero scheme $Z$, the blowup $\pi \text{\hspace{0.17em}⁣}:{\mathrm{Bl}}_{\mathcal{I}}X\to X$, and an $X$-scheme $f\text{\hspace{0.17em}⁣}:Y\to X$ such that $f^{-1}(Z)$ is an effective Cartier divisor on $Y$.

Choice. The Axiom of Choice is assumed, as in the statement; the cited suppliers used below are stated under it.

Universal property of an affine blowup chart: Let $\phi \text{\hspace{0.17em}⁣}:A\to B$ be a ring map, $I\subseteq A$ an ideal and $a\in I$, and suppose the image $b=\phi (a)$ is a nonzerodivisor in $B$ with $IB=bB$. Then there is a unique $A$-algebra homomorphism $A[I\mathrm{/}a]\to B$ sending $x\mathrm{/}{a}^n$ to the unique $y\in B$ with $x=b^{n}y$; equivalently, $\mathrm{Spec}B\to \mathrm{Spec}A[I\mathrm{/}a]$ is the unique $A$-morphism into the chart along which the image of $a$ generates $I{\mathcal{O}}_{\mathrm{Spec}B}$.

Affine blowup standard charts and overlaps: If $I=(f_0,\dots ,f_r)\subseteq A$ and $B_i=A[I\mathrm{/}{f}_i]$, the standard opens $U_i=\mathrm{Spec}{B}_i$ cover ${\mathrm{Bl}}_I\mathrm{Spec}A$, with transition maps sending $u_{ij}=(f_{j}t)\mathrm{/}(f_{i}t)$ to $u_{ji}^{-1}=f_j\mathrm{/}{f}_i$ on the overlaps.

Blowup of a scheme along an ideal sheaf: ${\mathrm{Bl}}_{\mathcal{I}}X={\mathrm{Proj}}_X\mathcal{R}(\mathcal{I})$ with structural morphism $\pi$, and the blowup is local on the base: over an affine open $\mathrm{Spec}A$ with $\mathcal{I}=(f_0,\dots ,f_r)$ it is covered by the charts $\mathrm{Spec}A[I\mathrm{/}{f}_i]$.

Scheme-theoretic inverse images of subschemes: For $f\text{\hspace{0.17em}⁣}:Y\to X$ and the closed subscheme $Z=V(\mathcal{I})$, the scheme-theoretic inverse image is $Y{\times }_{X}Z$, and the inverse-image ideal is $\mathrm{Im}(f^{\ast }\mathcal{I}\to {\mathcal{O}}_Y)$.

Effective cartier divisor: A Cartier divisor is effective when it has a local-equation representation by regular sections $f_i\in {\mathcal{O}}_X(U_i)$; the local principal ideals $f_i{\mathcal{O}}_{U_i}$ glue to an ideal sheaf, and a unit equation represents the empty divisor.

Morphisms of schemes are local on compatible open covers: Compatible morphisms on an open cover glue uniquely, and two morphisms out of $Y$ are equal if their restrictions to an open cover are equal.

Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains: The chart algebra $A[I\mathrm{/}a]$ is the degree-zero part of the localization of $R(I)$ at $a$, with $IA[I\mathrm{/}a]=aA[I\mathrm{/}a]$, and $a$ a nonzerodivisor; for $I=(a_0,\dots ,a_r)$, $a=a_0$, the chart receives a surjection $A[x_1,\dots ,x_r]\mathrm{/}(a{x}_i-a_i)\to A[I\mathrm{/}a]$.

The pulled-back center ideal is the relative twist; the exceptional divisor is Cartier: The inverse-image center ideal on the blowup is invertible and locally generated by a nonzerodivisor; its zero scheme is an effective Cartier divisor.