Finite birational algebras descend across a flat completion neighbourhood

Put $Q=C^{\mathrm{\prime }}\mathrm{/}C$ and choose $n\ge 1$ with $I^{n}Q=0$. Since $C^{\mathrm{\prime }}$ is finite over $C$ the quotient $Q$ is a finite $C$-module killed by $I^{n}C$, so it is a finite module over $C\mathrm{/}{I}^{n}C\cong B\mathrm{/}{I}^n$ and hence a finite $B$-module.

Because $Q$ is a $B\mathrm{/}{I}^n$-module, $Q{\otimes }_{B}C\cong Q{\otimes }_{B\mathrm{/}{I}^n}(C\mathrm{/}{I}^{n}C)\cong Q{\otimes }_{B\mathrm{/}{I}^n}(B\mathrm{/}{I}^n)=Q$, using the given identification $B\mathrm{/}{I}^n\cong C\mathrm{/}{I}^{n}C$; faithfully flat base change also makes $Q\to Q{\otimes }_{B}C$ injective.

Since $B$ is Noetherian and $Q$ is finite, $Q$ is Noetherian, so a finite free presentation can be extended indefinitely: all syzygies of $Q$ are finitely generated, and $Q$ admits a projective resolution $F_{\bullet }\to Q\to 0$ with every $F_i$ a finite free $B$-module.

Base change of the resolution along $B\to C$ computes the Ext of $Q$ over $C$, because $C$ is flat over $B$ and Hom of a finite free module commutes with the base change: ${\mathrm{Ext}}_B^1(Q,B){\otimes }_{B}C\cong {\mathrm{Ext}}_C^1(Q{\otimes }_{B}C,C)={\mathrm{Ext}}_C^1(Q,C)$, with $F_{\bullet }{\otimes }_{B}C$ a resolution of $Q$ by finite free $C$-modules.

Every element of $I^n$ kills ${\mathrm{Ext}}_B^1(Q,B)$: multiplication by $x\in I^n$ is zero on $Q$, so on the resolution it is a chain map lifting the zero map and, by successive projective lifts, is chain homotopic to zero, hence induces zero on the cohomology of the Hom complex.

Since $I^n$ kills ${\mathrm{Ext}}_B^1(Q,B)$, the same argument as for $Q$ shows ${\mathrm{Ext}}_B^1(Q,B){\otimes }_{B}C\cong {\mathrm{Ext}}_B^1(Q,B)$; combined with the previous step the base-change map ${\mathrm{Ext}}_B^1(Q,B)\to {\mathrm{Ext}}_C^1(Q,C)$ is an isomorphism.

The extension class of $0\to C\to C^{\mathrm{\prime }}\to Q\to 0$ in ${\mathrm{Ext}}_C^1(Q,C)$ therefore has a unique preimage in ${\mathrm{Ext}}_B^1(Q,B)$, which is the class of a short exact sequence $0\to B\to E\to Q\to 0$ of $B$-modules whose base change to $C$ is isomorphic to $0\to C\to C^{\mathrm{\prime }}\to Q\to 0$. The module $E$ is finite over $B$ because $B$ and $Q$ are.

Choose $0\mathrm{\ne }a\in I$. Since $Q$ is killed by $I^n$, we have $Q_a=0$, so localizing the extension at $a$ identifies $E_a$ with $B_a$. The kernel $K$ of $E\to E_a$ is killed by a power of $a$: it is a submodule of the finite $B$-module $E$, hence finite because $B$ is Noetherian, and each of its finitely many generators is killed by some power of $a$. After tensoring with the flat ring map $B\to C$, this kernel is the kernel of $C^{\mathrm{\prime }}\to C_a^{\mathrm{\prime }}$, which is zero because $C^{\mathrm{\prime }}\subset \mathrm{Frac}C$ is a domain and localization is injective. Faithful-flatness detects zero modules here: for any nonzero module element $m$, its cyclic submodule is $B\mathrm{/}J$ for a proper annihilator $J$; flatness embeds $(B\mathrm{/}J){\otimes }_{B}C=C\mathrm{/}JC$ into the tensor of the module, and [F4] makes $C\mathrm{/}JC$ nonzero. Thus $K=0$, and $E$ embeds as a $B$-submodule of $B_a\subset \mathrm{Frac}B$.

Let $M:=B_a\mathrm{/}E$. Flatness identifies $M{\otimes }_{B}C$ with the quotient of $C_a=B_a{\otimes }_{B}C$ by the image of $E{\otimes }_{B}C$. The extension isomorphism $E{\otimes }_{B}C\cong C^{\mathrm{\prime }}$ respects the submodule $C$; after localizing at $a$, both middle terms identify with $C_a$, and the isomorphism is the identity on this submodule. Therefore the image of $E{\otimes }_{B}C$ in $C_a$ is exactly $C^{\mathrm{\prime }}$. For $u,v\in E$, the class $m:=uv+E$ maps in $M{\otimes }_{B}C\cong C_a\mathrm{/}{C}^{\mathrm{\prime }}$ to zero, since the images of $u$ and $v$ lie in the ring $C^{\mathrm{\prime }}$. The faithful-flat detection argument in step 8.1 shows that $M\to M{\otimes }_{B}C$ is injective, so $m=0$ and $uv\in E$. With this multiplication $E=:B^{\mathrm{\prime }}$ is a ring, finite over $B$, with $B\subset B^{\mathrm{\prime }}\subset \mathrm{Frac}B$, and the extension isomorphism $E{\otimes }_{B}C\cong C^{\mathrm{\prime }}$ is an isomorphism of $C$-algebras.

Let $\mathfrak{p}\mathrm{\notin }V(I)$. Choose $b\in I\setminus \mathfrak{p}$. Since $I^{n}Q=0$, localization at $b$ gives $Q_b=0$, so the exact sequence in step 7.1 makes $B_b\to E_b$ an isomorphism of modules. It is already a ring map, and both rings have the multiplication inherited from $E\subseteq B_a$, so it is an isomorphism of rings. These principal opens cover $\mathrm{Spec}(B)\setminus V(I)$, proving that $B^{\mathrm{\prime }}$ agrees with $B$ away from $V(I)$. The algebra is birational because $B\subseteq B^{\mathrm{\prime }}\subseteq \mathrm{Frac}(B)$. AC and DC are inherited from the Ext-one classification and resolution suppliers; no formal-gluing theorem is used.