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

22 CHRISTIAN KAPPEN(i) For any semi-affinoid subdomain U of X, the uniformly rigid G-topology on X restricts to the uniformly rigid G-topology on U.(ii) If U ⊆ X is a finite union of retrocompact semi-affinoid subdomainsin X, then U is T urig -admissible, and every finite covering of U byretrocompact semi-affinoid subdomains in X is T urig -admissible.Proof. By Corollary 2.26 (i), the semi-affinoid subdomains in U are thesemi-affinoid subdomains in X contained in U, and by Corollary 2.23 (iii)the semi-affinoid morphisms to X with image in U correspond to the semiaffinoidmorphisms to U. Hence, statement (i) follows from Proposition2.33 (i) and (ii).To prove the second statement, let (U i ) i∈I be a finite family of retrocompactsemi-affinoid subdomains of X such that U is the union of the U i . Let Ybe any semi-affinoid K-space, and let ϕ: Y → X be any semi-affinoid morphismwhose image is lies in U. Then (ϕ −1 (U i )) i∈I is a retrocompact coveringof Y ; by Propostion 2.31, it admits a leaflike refinement. By Proposition2.33 (i), we conclude that U is a T urig -admissible subset of X, and by Proposition2.33 (ii) we see that the covering (U i ) i∈I of U is T urig -admissible. □In particular, Corollary 2.34 (ii) and the theorem of Gerritzen and Grauert[5] 7.3.5/1 show that if A is an affinoid K-algebra and if U ⊆ Sp A is anaffinoid subdomain, then U ⊆ sSp A is T urig -admissible.Remark 2.35 (quasi-compactness). Proposition 2.33 (ii) shows that semiaffinoidK-spaces are quasi-compact in T urig , cf. [5] p. 337. By the maximumprinciple for affinoid K-algebras, it follows that sSp (R[[S]] ⊗ R K) has noT urig -admissible covering by semi-affinoid subdomains whose rings of functionsare affinoid. In particular, the covering of sSp (R[[S]] ⊗ R K) providedby Berthelot’s construction is not T urig -admissible.Remark 2.36 (bases for T urig ). Proposition 2.33 implies that the semi-affinoidsubdomains form a basis for the uniformly rigid G-topology on a semiaffinoidK-space, cf. [5] p. 337. The retrocompact semi-affinoid subdomainsin sSp (K〈S〉) do not form a basis for T urig : Indeed, sSp (R[[S]] ⊗ R K) is asemi-affinoid subdomain in sSp (K〈S〉); by Lemma 2.27 and Remark 2.35,it does not admit a T urig -admissible covering by retrocompact semi-affinoidsubdomains in sSp (K〈S〉). Thus we see that even though the K-algebraK〈S〉 is affinoid, the uniformly rigid G-topology on sSp (K〈S〉) turns outto be strictly coarser than the rigid G-topology on Sp (K〈S〉).Remark 2.37 (Zariski topology). The Zariski topology T Zar on a semi-affinoidK-space X is generated by the non-vanishing loci D(f) of the semi-affinoidfunctions f on X. Using Proposition 2.31, Proposition 2.33 and the argumentsin [5] 9.1.4/7, one easily shows that T urig is finer than T Zar . Moreover,one shows that if U = ⋃ ni=1 D(f i) for semi-affinoid functions f i onX, then U is T urig -admissibly covered by the U ≥ε with ε ∈ √ |K|, whereU ≥ε = ⋃ ni=1 {x ; |f i(x)| ≥ ε}.