The only goal in this section is to prove the following lemma which will play a key role in algebraization of rig-étale algebras. We will use a bit of the theory of algebraic spaces to prove this lemma; an earlier version of this chapter gave a (much longer) proof using algebra and a bit of deformation theory that the interested reader can find in the history of the Stacks project.
\[ \xymatrix{ C \ar[r] & C/JC \\ B \ar[u] \ar[r] & B_0 \ar[u] \\ A \ar[r] \ar[u] & A/J \ar[u] } \]
with $A \to B$ of finite type, $B/JB = B_0$, $B \to C$ étale, and $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ étale over $\mathop{\mathrm{Spec}}(A) \setminus V(I)$.
Proof.
Set $X = \mathop{\mathrm{Spec}}(A)$, $X_0 = \mathop{\mathrm{Spec}}(A_0)$, $Y_0 = \mathop{\mathrm{Spec}}(B_0)$, $Z = \mathop{\mathrm{Spec}}(C)$, $Z_0 = \mathop{\mathrm{Spec}}(C_0)$. Furthermore, denote $U \subset X$, $U_0 \subset X_0$, $V_0 \subset Y_0$, $W \subset Z$, $W_0 \subset Z_0$ the complement of the vanishing set of $I$. Here is a picture to help visualize the situation:
\[ \xymatrix{ Z \ar[dd] & Z_0 \ar[l] \ar[d] \\ & Y_0 \ar[d] \\ X & X_0 \ar[l] } \quad \quad \quad \xymatrix{ W \ar[dd] & W_0 \ar[l] \ar[d] \\ & V_0 \ar[d] \\ U & U_0 \ar[l] } \]
The conditions in the lemma guarantee that
\[ \xymatrix{ W_0 \ar[r] \ar[d] & Z_0 \ar[d] \\ V_0 \ar[r] & Y_0 } \]
is an elementary distinguished square, see Derived Categories of Spaces, Definition 75.9.1. In addition we know that $W_0 \to U_0$ and $V_0 \to U_0$ are étale. The morphism $X_0 \subset X$ is a finite order thickening as $J$ is assumed nilpotent. By the topological invariance of the étale site we can find a unique étale morphism $V \to X$ of schemes with $V_0 = V \times _ X X_0$ and we can lift the given morphism $W_0 \to V_0$ to a unique morphism $W \to V$ over $X$. See Étale Morphisms, Theorem 41.15.2. Since $W_0 \to V_0$ is separated, the morphism $W \to V$ is separated too, see for example More on Morphisms, Lemma 37.10.3. By Pushouts of Spaces, Lemma 81.9.2 we can construct an elementary distinguished square
\[ \xymatrix{ W \ar[r] \ar[d] & Z \ar[d] \\ V \ar[r] & Y } \]
in the category of algebraic spaces over $X$. Since the base change of an elementary distinguished square is an elementary distinguished square (Derived Categories of Spaces, Lemma 75.9.2) we see that
\[ \xymatrix{ W_0 \ar[r] \ar[d] & Z_0 \ar[d] \\ V_0 \ar[r] & Y \times _ X X_0 } \]
is an elementary distinguished square. It follows that there is a unique isomorphism $Y \times _ X X_0 = Y_0$ compatible with the two squares involving these spaces because elementary distinguished squares are pushouts (Pushouts of Spaces, Lemma 81.9.1). It follows that $Y$ is affine by Limits of Spaces, Proposition 70.15.2. Write $Y = \mathop{\mathrm{Spec}}(B)$. It is clear that $B$ fits into the desired diagram and satisfies all the properties required of it.
$\square$
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