Invariance of the blowup under invertible (fractional) rescaling of the ideal

Assume the Axiom of Choice. Let $X$ be integral, let $\mathcal{I}\subseteq {\mathcal{O}}_X$ be a quasi-coherent ideal of finite type, and let $J\subseteq {\mathcal{K}}_X$ be an invertible fractional ideal. Its product $F=J\mathcal{I}$ is a subsheaf of ${\mathcal{K}}_X$; it need not be contained in ${\mathcal{O}}_X$. Define its fractional Rees algebra and blowup by $\mathcal{R}(F)={⨁}_{n\ge 0}{F}^n,F^0={\mathcal{O}}_X,{\mathrm{Bl}}_{F}X={\mathrm{Proj}}_X\mathcal{R}(F)\mathrm{.}$ Here $F^n\cong J^{\otimes n}\otimes {\mathcal{I}}^n$, with multiplication induced inside ${\mathcal{K}}_X$, so the algebra is quasi-coherent and generated in degree one. When $F\subseteq {\mathcal{O}}_X$, this is the ordinary ideal blowup of Blowup of a scheme along an ideal sheaf.

On a nonempty affine open $U=\mathrm{Spec}A$ trivializing $J$, choose $g\in \mathrm{Frac}(A{)}^{\times }$ with $J{\mathrm{\mid }}_U=gA$. Multiplication by $g^n$ is an $A$-module isomorphism $I^n\to (gI{)}^n$ for every $n$, with inverse division by $g^n$. These maps respect multiplication and give a graded algebra isomorphism $R(I)\to R(gI)$. The inverse of its contravariantly induced Proj map defines ${\rho }_{\mathcal{I},J}{\mathrm{\mid }}_U:{\mathrm{Bl}}_{\mathcal{I}}U⟶{\mathrm{Bl}}_{J\mathcal{I}}U\mathrm{.}$ No assertion that $g\in A$ is needed.

Replacing $g$ by $ug$, $u\in A^{\times }$, changes the degree-$n$ map by $u^n$. This automorphism induces the identity on Proj: on any homogeneous localization, numerator and denominator of a degree-zero fraction acquire the same power of $u$, which cancels. Thus the local maps agree on overlaps and glue to a canonical isomorphism of $X$-schemes ${\rho }_{\mathcal{I},J}:{\mathrm{Bl}}_{\mathcal{I}}X\stackrel{\sim }{\to }{\mathrm{Bl}}_{J\mathcal{I}}X\mathrm{.}$ Division by $g^n$ gives its inverse, including when the original ideal is zero and both blowups are empty. This construction uses relative Proj and is compatible with restriction to opens.

An invertible sheaf $L$ on an integral scheme can be realized as an invertible fractional ideal by choosing a nonzero basis of its generic fiber: local sections inject into that fiber, since locally $L$ is free and the coordinate rings are domains. Consequently the algebra ${⨁}_{n\ge 0}{\mathcal{I}}^n\otimes L^{\otimes n}$ has the same relative Proj as $\mathcal{R}(\mathcal{I})$. In particular one may use an ample twist that makes $\mathcal{I}\otimes L$ globally generated, without treating that sheaf as an ordinary ideal. For $J=(1\mathrm{/}x)$ and $\mathcal{I}=\mathcal{O}$ on an affine domain containing a nonunit $x$, the product is fractional, illustrating why the distinction is necessary.

The ordinary effective Cartier rescaling also works without integrality of $X$. For any scheme $X$, a quasi-coherent ideal $\mathcal{I}$ and an effective Cartier divisor with ideal $J$ (Effective cartier divisor), use here the Rees Proj ${\mathrm{Proj}}_X({⨁}_{n\ge 0}{\mathcal{I}}^n)$ even if $\mathcal{I}$ is not of finite type: ideal powers commute with affine localization, so this is a quasi-coherent graded algebra to which Relative Proj of a graded quasi-coherent algebra applies. Write $J=g\mathcal{O}$ locally, where $g$ is a nonzerodivisor. Multiplication by $g^n$ is an isomorphism $I^n\to g^n{I}^n$ in every degree, with inverse on its image, so it gives a graded Rees algebra isomorphism and a local blowup isomorphism. On overlaps $g$ changes by a unit; the same degree-zero cancellation proves that these isomorphisms glue canonically to ${\mathrm{Bl}}_{\mathcal{I}}X\cong {\mathrm{Bl}}_{J\mathcal{I}}X$. Thus multiplying the center ideal by an effective Cartier ideal leaves the blowup scheme canonically unchanged on arbitrary $X$, including $I=0$. This assertion concerns the blowup object; the center and the open complement used to define a strict transform may change.