The Stacks project

85.5 Rig-étale morphisms

We can use the notion introduced in the previous section to define a new type of morphism of locally Noetherian formal algebraic spaces. Before we do so, we have to check it is a local property.

Lemma 85.5.1. For morphisms $A \to B$ of the category $\textit{WAdm}^{Noeth}$ (Formal Spaces, Section 84.16) consider the condition $P=$“for some ideal of definition $I$ of $A$ the topology on $B$ is the $I$-adic topology, the ring map $A/I \to B/IB$ is of finite type and $A \to B$ satisfies the equivalent conditions of Lemma 85.4.1”. Then $P$ is a local property, see Formal Spaces, Remark 84.16.5.

Proof. We have to show that Formal Spaces, Axioms (1), (2), and (3) hold for maps between Noetherian adic rings. For a Noetherian adic ring $A$ with ideal of definition $I$ we have $A\{ x_1, \ldots , x_ r\} = A[x_1, \ldots , x_ r]^\wedge $ as topological $A$-algebras (see Formal Spaces, Remark 84.21.2). We will use without further mention that we know the axioms hold for the property “$B$ is a quotient of $A[x_1, \ldots , x_ r]^\wedge $”, see Formal Spaces, Lemma 84.22.6.

Let a diagram as in Formal Spaces, Diagram (84.16.2.1) be given with $A$ and $B$ in the category $\textit{WAdm}^{Noeth}$. Pick an ideal of definition $I \subset A$. By the remarks above the topology on each ring in the diagram is the $I$-adic topology. Since $A \to A'$ and $B \to B'$ are étale we see that $\mathop{N\! L}\nolimits ^\wedge _{(A')^\wedge /A}$ and $\mathop{N\! L}\nolimits ^\wedge _{(B')^\wedge /B}$ are zero. By Lemmas 85.3.2 and 85.3.3 we get

\[ H^ i(\mathop{N\! L}\nolimits ^\wedge _{(B')^\wedge /(A')^\wedge }) \cong H^ i(\mathop{N\! L}\nolimits ^\wedge _{(B')^\wedge /A}) \quad \text{and}\quad H^ i(\mathop{N\! L}\nolimits ^\wedge _{B/A} \otimes _ B (B')^\wedge ) \cong H^ i(\mathop{N\! L}\nolimits ^\wedge _{(B')^\wedge /A}) \]

for $i = -1, 0$. Since $B$ is Noetherian the ring map $B \to B' \to (B')^\wedge $ is flat (Algebra, Lemma 10.96.2) hence the tensor product comes out. Moreover, as $B$ is $I$-adically complete, then if $B \to B'$ is faithfully flat, so is $B \to (B')^\wedge $. From these observations Formal Spaces, Axioms (1) and (2) follow immediately.

We omit the proof of Formal Spaces, Axiom (3). $\square$

Definition 85.5.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally Noetherian formal algebraic spaces over $S$. We say $f$ is rig-étale if $f$ satisfies the equivalent conditions of Formal Spaces, Lemma 84.16.3 (in the setting of locally Noetherian formal algebraic spaces, see Formal Spaces, Remark 84.16.4) for the property $P$ of Lemma 85.5.1.

To be sure, a rig-étale morphism is locally of finite type.

Lemma 85.5.3. A rig-étale morphism of locally Noetherian formal algebraic spaces is locally of finite type.

Proof. The property $P$ in Lemma 85.5.1 implies the equivalent conditions (a), (b), (c), and (d) in Formal Spaces, Lemma 84.22.6. Hence this follows from Formal Spaces, Lemma 84.22.9. $\square$


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