Flat base change for blowups, and failure without flatness

Flat pullback commutes with the ideal powers and with the inverse image ideal: the natural maps $g^{\ast }\mathcal{I}\to g^{-1}\mathcal{I}$ and $g^{\ast }({\mathcal{I}}^n)\to (g^{-1}\mathcal{I}{)}^n$ are isomorphisms. Affine-locally over $U=\mathrm{Spec}A\subseteq X$ with $\mathcal{I}{\mathrm{\mid }}_U=\tilde{I}$ and $U^{\mathrm{\prime }}=\mathrm{Spec}B\subseteq X^{\mathrm{\prime }}$ mapping into $U$, flatness of $g$ says $B$ is a flat $A$-module; the sequence $0\to I^n\to A\to A\mathrm{/}{I}^n\to 0$ then stays exact after $-{\otimes }_{A}B$, so $B{\otimes }_A{I}^n\to B$ is injective, and its image is the ideal $I^{n}B=(IB{)}^n$; the pullback sheaf $g^{\ast }({\mathcal{I}}^n)$ restricts to $\widetilde{(B{\otimes }_A{I}^n)}$ by [F2] and $(g^{-1}\mathcal{I}{)}^n$ restricts to $\widetilde{(IB{)}^n}$ by [F2] and [F3], so the comparison is an isomorphism, and taking $n=1$ identifies $g^{\ast }\mathcal{I}$ with its image $g^{-1}\mathcal{I}$ in ${\mathcal{O}}_{X^{\mathrm{\prime }}}$.

Consequently $g^{\ast }\mathcal{R}(\mathcal{I})\cong \mathcal{R}(g^{-1}\mathcal{I})$ as quasi-coherent graded ${\mathcal{O}}_{X^{\mathrm{\prime }}}$-algebras: by step 1.1 the degree-$n$ pieces are both $(g^{-1}\mathcal{I}{)}^n$, and the pullback of the multiplication ${\mathcal{I}}^m\otimes {\mathcal{I}}^n\to {\mathcal{I}}^{m+n}$ is the multiplication of the inverse image ideal, so the graded algebra structures agree; all pieces are quasi-coherent by [F2], [F3] and [F4].

Applying [F6] to the morphism $g\text{\hspace{0.17em}⁣}:X^{\mathrm{\prime }}\to X$ and the graded algebra $\mathcal{A}=\mathcal{R}(\mathcal{I})$ gives a canonical isomorphism of $X^{\mathrm{\prime }}$-schemes ${\mathrm{Bl}}_{\mathcal{I}}X{\times }_X{X}^{\mathrm{\prime }}={\mathrm{Proj}}_X\mathcal{R}(\mathcal{I}){\times }_X{X}^{\mathrm{\prime }}\cong {\mathrm{Proj}}_{X^{\mathrm{\prime }}}{g}^{\ast }\mathcal{R}(\mathcal{I})$, and step 2.1 identifies the target with ${\mathrm{Proj}}_{X^{\mathrm{\prime }}}\mathcal{R}(g^{-1}\mathcal{I})={\mathrm{Bl}}_{g^{-1}\mathcal{I}}{X}^{\mathrm{\prime }}$ by [F5]; the inverse of this composite is the canonical isomorphism of the statement. The comparison is compatible with the structural morphisms because both sides are the relative Proj of the pulled-back graded algebra over $X^{\mathrm{\prime }}$, and with the relative twists by the corresponding clause of [F6].

The comparison need not be an isomorphism without flatness. Take $A=k[x,y]$, $I=(x,y)$ and $B=A\mathrm{/}(y)=k[x]$. Then $IB=(x)$ is principal regular and its blowup is $\mathrm{Spec}B$. Base changing the two charts of the original blowup gives $k[x,T]\mathrm{/}(xT)$ and $k[U]$, respectively, with inverse ratio gluing. The fiber of this base-changed blowup over $x=0$ is the two affine lines glued by $U=T^{-1}$, hence ${\mathbb{P}}_k^1$, whereas the fiber of ${\mathrm{Bl}}_{(x)}\mathrm{Spec}B$ is $\mathrm{Spec}k$. Thus the comparison is not an isomorphism. The difference already appears in degree two: $B{\otimes }_A{I}^2=I^2\mathrm{/}y{I}^2$ contains the nonzero class of $xy$, since $x\mathrm{\notin }{I}^2$ and cancellation of $y$ in $A$ shows $xy\mathrm{\notin }y{I}^2$. This class maps to zero in $(IB{)}^2$, and is killed by $x$ since $x^{2}y\in y{I}^2$. Hence the flatness hypothesis cannot be dropped, and the stated torsion caveat is proved within this item.