UNIFORMLY RIGID SPACES 1. Introduction Let K be a non ...

16 CHRISTIAN KAPPENthe resulting squareY ′ ✲ X ′❄Y ✲ ❄Xis cartesian in the category of flat formal R-schemes of locally ff type.Proof. The universal property of the fibered product in the category of flatformal R-schemes of locally ff type is readily verified using the universalproperty of admissible blowups, the fact that the functor rig maps admissibleblowups to isomorphisms and the fact that rig is faithful on the categoryof flat formal R-schemes of locally ff type.□In the following, we write × ′ to denote the fibered product in the categoryof flat formal R-schemes of locally ff type. It is obtained from the usualfibered product by dividing out the coherent ideal of π-torsion; in particular,fibered products of affine flat formal R-schemes of ff type in the categoryof flat formal R-schemes of locally ff type are again affine.As we have just observed, admissible blowups of flat formal R-schemes arepreserved under pullback in the category of flat formal R-schemes of locallyff type. The same is true for open immersions and completion morphisms,since they are flat and since they are preserved under pullback in the categoryof all formal R-schemes of locally ff type.Corollary 2.23. Let X be a semi-affinoid K-space, let U ⊆ X be a semiaffinoidsubdomain, and let Y → X be a morphism of semi-affinoid K-spaces.(i) The preimage of U in Y is a semi-affinoid subdomain in Y .(ii) If U → X represents U as a semi-affinoid subdomain in X and ifY → X is a model of Y → X, then the projection U × ′ X Y → Yrepresents the preimage of U as a semi-affinoid subdomain in Y .(iii) Semi-affinoid subdomains are semi-affinoid pre-subdomains, and ifϕ represents U as a semi-affinoid subdomain in X, then ϕ urig representsall semi-affinoid morphisms to X with image in U.The analogous statements hold if we consider retrocompact or elementarysemi-affinoid subdomains and their retrocompact or elementary representations.Proof. Statement (ii) implies statement (i) in view of Corollary 2.13 (iv).To show (ii), let us consider a factorizationU −→ ϕn ϕ n−1 ϕ 1 ϕ 0U n −→ · · · −→ U1 −→ X(∗)