Lemma 88.22.2. Let $X$ be a locally Noetherian formal algebraic space which is locally of finite type over a complete discrete valuation ring $A$. Let $X' \subset X$ be as in Lemma 88.22.1. If $X \to X \times _{\text{Spf}(A)} X$ is rig-étale and rig-surjective, then $X' = \text{Spf}(A)$ or $X' = \emptyset $.
Proof. (Aside: the diagonal is always locally of finite type by Formal Spaces, Lemma 87.15.5 and $X \times _{\text{Spf}(A)} X$ is locally Noetherian by Formal Spaces, Lemmas 87.24.4 and 87.24.8. Thus imposing the conditions on the diagonal morphism makes sense.) The diagram
\[ \xymatrix{ X' \ar[r] \ar[d] & X' \times _{\text{Spf}(A)} X' \ar[d] \\ X \ar[r] & X \times _{\text{Spf}(A)} X } \]
is cartesian. Hence $X' \to X' \times _{\text{Spf}(A)} X'$ is rig-étale and rig-surjective by Lemma 88.21.4. Choose an affine formal algebraic space $U$ and a morphism $U \to X'$ which is representable by algebraic spaces and étale. Then $U = \text{Spf}(B)$ where $B$ is an adic Noetherian topological ring which is a flat $A$-algebra, whose topology is the $\pi $-adic topology where $\pi \in A$ is a uniformizer, and such that $A/\pi ^ n A \to B/\pi ^ n B$ is of finite type for each $n$. For later use, we remark that this in particular implies: if $B \not= 0$, then the map $\text{Spf}(B) \to \text{Spf}(A)$ is a surjection of sheaves (please recall that we are using the fppf topology as always). Repeating the argument above, we see that
\[ W = U \times _{X'} U = X' \times _{X' \times _{\text{Spf}(A)} X'} (U \times _{\text{Spf}(A)} U) \longrightarrow U \times _{\text{Spf}(A)} U \]
is a closed immersion and rig-étale and rig-surjective. We have $U \times _{\text{Spf}(A)} U = \text{Spf}(B \widehat{\otimes }_ A B)$ by Formal Spaces, Lemma 87.16.4. Then $B \widehat{\otimes }_ A B$ is a flat $A$-algebra as the $\pi $-adic completion of the flat $A$-algebra $B \otimes _ A B$. Hence $W = U \times _{\text{Spf}(A)} U$ by Lemma 88.21.13. In other words, we have $U \times _{X'} U = U \times _{\text{Spf}(A)} U$ which in turn means that the image of $U \to X'$ (as a map of sheaves) maps injectively to $\text{Spf}(A)$. Choose a covering $\{ U_ i \to X'\} $ as in Formal Spaces, Definition 87.11.1. In particular $\coprod U_ i \to X'$ is a surjection of sheaves. By applying the above to $U_ i \coprod U_ j \to X'$ (using the fact that $U_ i \amalg U_ j$ is an affine formal algebraic space as well) we see that $X' \to \text{Spf}(A)$ is an injective map of fppf sheaves. Since $X'$ is flat over $A$, either $X'$ is empty (if $U_ i$ is empty for all $i$) or the map is an isomorphism (if $U_ i$ is nonempty for some $i$ when we have seen that $U_ i \to \text{Spf}(A)$ is a surjective map of sheaves) and the proof is complete. $\square$
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)