The Stacks project

87.9 A pushout argument

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.

Lemma 87.9.1. Let $A$ be a Noetherian ring and $I \subset A$ an ideal. Let $J \subset A$ be a nilpotent ideal. Consider a commutative diagram

\[ \xymatrix{ C \ar[r] & C_0 \ar@{=}[r] & C/JC \\ & B_0 \ar[u] \\ A \ar[r] \ar[uu] & A_0 \ar[u] \ar@{=}[r] & A/J } \]

whose vertical arrows are of finite type such that

  1. $\mathop{\mathrm{Spec}}(C) \to \mathop{\mathrm{Spec}}(A)$ is étale over $\mathop{\mathrm{Spec}}(A) \setminus V(I)$,

  2. $\mathop{\mathrm{Spec}}(B_0) \to \mathop{\mathrm{Spec}}(A_0)$ is étale over $\mathop{\mathrm{Spec}}(A_0) \setminus V(IA_0)$, and

  3. $B_0 \to C_0$ is étale and induces an isomorphism $B_0/IB_0 = C_0/IC_0$.

Then we can fill in the diagram above to a commutative diagram

\[ \xymatrix{ C \ar[r] & C/JC \\ B \ar[u] \ar[r] & B_0 \ar[u] \\ A \ar[r] \ar[u] & A/J \ar[u] } \]

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 74.9.1. 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 80.9.2 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 74.9.2) 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 80.9.1). It follows that $Y$ is affine by Limits of Spaces, Proposition 69.15.2. 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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