The blowup is independent of chosen ideal generators

The Rees algebra $R(I)={⨁}_{n\ge 0}{I}^n{t}^n$ and the affine blowup algebra $A[I\mathrm{/}a]=(R(I){)}_{(a)}$ for $a\in I$ depend only on $I$ and $a$: by [F2] the affine blowup algebra is independent of the chosen generating set and of the representative of the homogeneous localisation, up to canonical isomorphism.

Let $U=\mathrm{Spec}A$ be an affine open and let $f_0,\dots ,f_r$ generate $I=\mathrm{\Gamma }(U,\mathcal{I})$. By [F3] the standard opens $U_i=D_{+}(f_{i}t)=\mathrm{Spec}A[I\mathrm{/}{f}_i]$ cover ${\mathrm{Bl}}_{I}U$, with overlaps $U_i\cap U_j=D(u_{ij})$ and the canonical identifications $(B_i{)}_{u_{ij}}=(B_j{)}_{u_{ji}}$ of chart rings given by the ratios of the degree-one elements $f_{i}t,f_{j}t$ of $R(I)$.

Now let $g_0,\dots ,g_s$ be a second finite family generating the same ideal $I$. Both families present open covers of the single scheme ${\mathrm{Proj}}_{U}R(I)={\mathrm{Bl}}_{I}U$ by [F1]: a standard chart $D_{+}(ht)$ is the basic open of the degree-one element $ht\in R(I{)}_1$, so the charts of the two families are open subschemes of the same relative Proj, and every overlap $D_{+}(f_{i}t)\cap D_{+}(g_{j}t)$ is the basic open of the degree-zero ratio of the two degree-one elements inside this Proj, identified with the corresponding localised chart ring as in [F3].

If a chart occurs in both families, that is $f_i=g_j=h$ for some $h\in I$, the two chart rings are both the affine blowup algebra $A[I\mathrm{/}h]$ of [F2], and the identification is the identity of this common algebra, well defined independently of the family by step 1.1.

The chartwise identifications of steps 2.1 and 2.2 are the restrictions of the identity of the single scheme ${\mathrm{Bl}}_{I}U={\mathrm{Proj}}_{U}R(I)$ to the members and pairwise overlaps of the two covers, so they satisfy the identity and cocycle conditions automatically, and glue to an isomorphism of presentations of ${\mathrm{Bl}}_{I}U$; over an affine cover of $X$ these isomorphisms are compatible on overlaps by [F4], so they glue to a canonical isomorphism of $X$-schemes between the presentations of ${\mathrm{Bl}}_{\mathcal{I}}X$ built from the two generating families. In particular ${\mathrm{Bl}}_{\mathcal{I}}X$ does not depend on a chosen finite generating set, and the affine blowup presentations $A[I\mathrm{/}a]$, $a\in I$, are precisely the standard charts $D_{+}(at)$ of the canonical blowup.