Nonflat base change of a blowup can fail

The quotient $g\text{\hspace{0.17em}⁣}:A\to B=A\mathrm{/}(y)$ is not flat: the sequence $0\to A\stackrel{\cdot y}{\to }A\to B\to 0$ is exact because $y$ is a nonzerodivisor of the polynomial ring $A$, so if $B$ were flat over $A$ the sequence $0\to B\stackrel{\cdot y}{\to }B\to B\to 0$ would be exact by [F4]; but $y=0$ in $B$, so the first map is the zero map with kernel $B=k[x]\mathrm{\ne }0$, and injectivity of the first map would force $B=0$, a contradiction.

The comparison of Rees algebras is not an isomorphism. Its degree-two source is $I^2{\otimes }_{A}B=I^2\mathrm{/}y{I}^2$. The class of $xy$ is nonzero: if $xy=yh$ with $h\in I^2$, cancellation of $y$ in $A$ would give $x=h\in I^2$, a contradiction. It maps to zero in $(IB{)}^2=(x{)}^{2}B$ and is killed by $x$, since $x^{2}y\in y{I}^2$. Thus it is a nonzero torsion kernel class. The generators $x^2,xy,y^2$ are not an $A$-basis; no freeness of $I^2$ is used.

The source ${\mathrm{Bl}}_{(x)}\mathrm{Spec}B$ is $\mathrm{Spec}B$: the pullback ideal $(x)\subseteq k[x]$ is principal generated by the nonzerodivisor $x$, and by [F2] its single standard chart is $\mathrm{Spec}B[(x)\mathrm{/}x]=\mathrm{Spec}B$; since that chart covers the blowup, the blowup is the identity on $\mathrm{Spec}B$.

By relative Proj base change the target is $T={\mathrm{Proj}}_B(B{\otimes }_{A}R(I))$. The incidence presentation identifies it with $V(xv)\subset {\mathbb{P}}_B^1$. Its two components are the section $V(v)\cong \mathrm{Spec}B$ and the closed fiber $V(x)\cong {\mathbb{P}}_k^1$ over the origin. On $D_{+}(u)$ the ring is $k[x,T]\mathrm{/}(xT)$, with component ideals $(T)$ and $(x)$ and their intersection point $(x,T)$. The other chart is $k[U]$ with $x=0$, extending the latter affine-line piece to ${\mathbb{P}}_k^1$ and adding no further component. Thus the target has two irreducible components.

The canonical comparison sends the ratio $T=y\mathrm{/}x$ to zero on the first chart, so its image is the section $V(v)$. Its source is $\mathrm{Spec}B$ by step 1.3. The source fiber over $x=0$ is $\mathrm{Spec}k$, while the target fiber is ${\mathbb{P}}_k^1$ by step 1.4; hence this morphism over $\mathrm{Spec}B$ is not an isomorphism. This proves the failure without flatness, independently of any assertion that the degree-two generators form a basis.