# An example of a morphism of rigid analytic spaces with affinoid base which is proper but does not satisfy $(\dagger)$

Let $$k$$ be a complete non-archimedean field and let $$\varphi \colon X \to Y$$ be a morphism of rigid analytic spaces over $$k$$, where $$\newcommand{\Sp}{\operatorname{Sp}}Y = \Sp(B)$$ is affinoid. Consider the following condition:

$$(\dagger)$$ The morphism $$\varphi$$ is separated and there exist two finite admissable affinoid coverings $$\mathfrak{U} = (U_i)_{i \in I}$$, $$\mathfrak{V} = (V_i)_{i \in I}$$ of $$X$$ such that $$V_i \Subset_Y U_i$$ (i.e. $$V_i$$ lies relatively compact in $$U_i$$ w.r.t. $$Y$$) for all $$i \in I$$.

A morphism $$\varphi \colon X \to Y$$ (with $$Y$$ not necessarily affinoid) is called proper if it is separated and there is an admissable affinoid covering $$(W_j)_{j \in J}$$ of $$Y$$ such that for each $$j \in J$$ the morphism $$\varphi^{-1}(W_j) \to W_j$$ satisfies condition $$(\dagger)$$.

If $$Y$$ is affinoid, then obviously a morphism satisfying condition $$(\dagger)$$ is proper, whereas the converse is not clear. In fact, Bosch in his book Lectures of Formal and Rigid Geometry writes at the beginning of section 6.4 that $$(\dagger)$$ is slightly stronger than properness.

Is there an example of a proper morphism with affinoid base which does not satisfy condition $$(\dagger)$$?

These two notions are actually equivalent (at least if $$k$$ is the fraction field of a dvr $$R$$), but I do not know any direct way to see this.

The proof I know heavily uses the theory of formal schemes. The three main results we need are Lemma 2.5, Lemma 2.6 and the statement after Corollary 3.2 from Lutkebohmert's paper Formal-algebraic and rigid-analytic geometry''.

Lemma 2.5: Let $$\mathfrak W \to \mathfrak V=\text{Spf}(A)$$ be a morphism of admissible formal schemes whose generic fibers are affinoid and let $$\mathfrak U \subset \mathfrak W$$ be an open formal subscheme. Then $$\mathfrak U_K$$ is relatively compact in $$\mathfrak W_K$$ over $$\mathfrak V_K$$ if and only if the Zariski-closure $$\overline{\mathfrak U}_0$$ of $$\mathfrak U_0$$ in $$\mathfrak W_0$$ is proper over $$\mathfrak V_0$$.

Lemma 2.6: Let $$f\colon \mathfrak X \to \mathfrak Y$$ be a morphism of admissible formal schemes and let $$f_K\colon \mathfrak X_K \to \mathfrak Y_K$$ be its generic fiber. If the rigid map $$f_K$$ is proper, the formal map $$f$$ is proper.

Claim on page 350: Let $$f\colon \mathfrak X \to \mathfrak Y=\text{Spf} B$$ be a proper morphism of admissible formal schemes. Let $$\mathfrak U$$ be a formal open subscheme of $$\mathfrak X$$ which is affine. Then there exists an admissible blowing-up $$\mathfrak X \to \mathfrak X'$$ with a center contained in the complement of $$\mathfrak U_0$$ such that there exists an open subscheme $$\mathfrak U'$$ of $$\mathfrak X'$$ whose associated rigid subspace $$\mathfrak U'_K$$ is affinoid and such that the Zariski-closure $$\overline{\mathfrak U_0}$$ of $$\mathfrak U_0$$ in $$\mathfrak X'_0$$ is contained in $$\mathfrak U'_0$$.

These results easily imply the desired claim. Indeed, suppose that $$g\colon X \to \text{Sp}(A)$$ is a proper map. Choose some topologically finitely generated ring of definition $$A_0 \subset A$$. Then using the standard machinery on formal models, we can find a morphism of admissible formal schemes $$f\colon \mathfrak X \to \text{Spf } A_0$$ such that the generic fiber $$f_K$$ is equal to $$g$$. Then Lemma 2.6 reads that $$f$$ is proper as a map of formal schemes.

Now comes the key step: we use the last cited claim to find a good covering of $$X$$. Namely, we choose any covering of $$\mathfrak X$$ by open affine formal schemes $$\mathfrak U_i$$. We apply that claim to find an admissible blow-up $$\mathfrak X'_i \to \mathfrak X$$ with an open formal subscheme $$\mathfrak U'_i\subset X'_i$$ such that $$(\mathfrak U'_i)_K$$ is affinoid and $$\overline{\mathfrak U_0}$$ (that is proper as a closed subscheme of $$\mathfrak X'_0$$) is contained in $$(\mathfrak U'_i)_0$$. Lemma 2.5 implies that $$(\mathfrak U_i)_K$$ is relatively compact in $$(\mathfrak U'_i)_K$$. The last thing to observe is that $$(\mathfrak U_i)_K$$ form a finite covering of the rigid space $$X$$.

P.S. Lutkebohmert's paper is written entirely in the noetherian setup. So if one wants to give a proof along these lines without noetherianness assumption on $$\mathcal O_K$$, then one needs to reprove these results in the set-up of arbitrary (complete) rank-$$1$$ valuation rings. My recollection is that it should not be that difficult given the recent finiteness results of Fujiwara and Kato (Fujiwara, Kato $$$$Foundations of Rigid Geometry I'')

• Thank you very much. I already came to suspect this. This should also follow from [Temkin, On local properties of non-archimedean analytic spaces, Cor. 4.5] in complete generality. – Jakob Werner Oct 12 '19 at 12:32