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

UNIFORMLY RIGID SPACES 7of ff type if it admits a presentation as a quotient of a mixed formal powerseries ring, as above. If S and T are finite systems of variables and ifϕ : R[[S]]〈T 〉 → A is a surjection, then the ϕ-image of (S, T ) will be calleda formal generating system for A.Lemma 2.1. If A is a topological R-algebra of ff type, then the biggest idealof definition of A coincides with the Jacobson radical of A. Moreover, anyR-homomorphism of topological R-algebras of ff type is continuous.Proof. If A is a topological R-algebra of ff type, then the biggest ideal aof definition of A coincides with the Jacobson radical jac A of A. Indeed,a is contained in every maximal ideal of A since A is a-adically complete,and A/a is a Jacobson ring since it is of finite type over the residue field kof R. Hence, the topology on A is determined by the ring structure of A.Let now A → B be a homomorphism of R-algebras of ff type; by what wehave seen so far, it suffices to see that ϕ is continuous for the Jacobson-adictopologies. However, for any maximal ideal n ⊆ B, the preimage m := n∩Aof n in A is maximal, since k ⊆ A/m ⊆ B/n, where B/n is a finite fieldextension of k because the quotient B/jac B is of finite type over k. □In particular, the topology on A an be recovered from the ring structure onA, and the category of R-algebras of ff type is canonically equivalent to thecategory of topological R-algebras of ff type. Lemma 2.1 implies that thecategory of R-algebras of ff type admits amalgamated sums ˆ⊗.2.2. Semi-affinoid algebras. We define semi-affinoid K-algebras as thegeneric fibers of R-algebras of ff type, and we define the category of semiaffinoidK-spaces as the dual of the category of semi-affinoid K-algebras:Definition 2.2. Let A be a K-algebra.(i) An R-model of A is an R-subalgebra A ⊆ A such that the naturalhomomorphism A ⊗ R K → A is an isomorphism.(ii) The K-algebra A is called semi-affinoid if it admits an R-model offf type.(iii) A homomorphism of semi-affinoid K-algebras is a homomorphismof underlying K-algebras.(iv) The category of semi-affinoid K-spaces is the dual of the categoryof semi-affinoid K-algebras. If A is a semi-affinoid K-algebra, wewrite sSp A to denote the corresponding semi-affinoid K-space, andif ϕ : sSp B → sSp A is a morphism of semi-affinoid K-spaces, wewrite ϕ ∗ to denote the corresponding K-algebra homomorphism.There exists no general analog of the Noether normalization theorem forsemi-affinoid K-algebras, cf. [21] I.2.3.5. However, one readily verifies thatif A is a semi-affinoid K-algebra admitting a local R-model of ff type, then