Lemma 86.5.1. For morphisms $A \to B$ of the category $\textit{WAdm}^{Noeth}$ (Formal Spaces, Section 85.16) 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 86.4.1”. Then $P$ is a local property, see Formal Spaces, Remark 85.16.5.
86.5 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.
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 85.21.2). 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 85.22.6.
Let a diagram as in Formal Spaces, Diagram (85.16.2.1) 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 86.3.2 and 86.3.3 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 86.5.2. 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 85.16.3 (in the setting of locally Noetherian formal algebraic spaces, see Formal Spaces, Remark 85.16.4) for the property $P$ of Lemma 86.5.1.
To be sure, a rig-étale morphism is locally of finite type.
Lemma 86.5.3. A rig-étale morphism of locally Noetherian formal algebraic spaces is locally of finite type.
Proof. The property $P$ in Lemma 86.5.1 implies the equivalent conditions (a), (b), (c), and (d) in Formal Spaces, Lemma 85.22.6. Hence this follows from Formal Spaces, Lemma 85.22.9. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)