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

UNIFORMLY RIGID SPACES 13for i = 1, 2, where σ i : B i → B 1 ˆ⊗ A B 2 is the ith coprojection. Settingτ := τ ⊗ R K and σ i := σ i ⊗ R K, we obtain τ i = τ ◦ σ i for i = 1, 2. We mustshow that τ is uniquely determined by this property. Letτ ′ : (B 1 ˆ⊗ A B 2 ) ⊗ R K → Cbe any K-homomorphism satisfying τ i = τ ′ ◦ σ i for i = 1, 2. By Corollary2.13 (iv), there exists an R-model C ′ of ff type for C containing C suchthat τ ′ restricts to an R-morphism τ ′ : B 1 ˆ⊗ A B 2 → C ′ ; then τ ′ = τ ′ ⊗ R K.It suffices to show that τ ′ coincides with τ composed with the inclusionι : C ⊆ C ′ . For i = 1, 2, the compositions τ ′ ◦ σ i and ι ◦ τ ◦ σ i coincide afterinverting π, hence they coincide because π is not a zero divisor in B i , fori = 1, 2. The universal property of (B 1 ˆ⊗ A B 2 , σ 1 , σ 2 ) implies that τ ′ = ι ◦ τ,as desired.□Passing to the opposite category, we see that the category of semi-affinoidK-spaces has fibered products.2.2.5. The Nullstellensatz.Proposition 2.16. Semi-affinoid K-algebras are Jacobson rings.Proof. Any quotient of a semi-affinoid K-algebra is again semi-affinoid;hence it suffices to show that if A is a semi-affinoid K-algebra and if f ∈ Ais a semi-affinoid function such that f(x) = 0 for all x ∈ sSp A, then f isnilpotent. We may divide A by its nilradical and thereby assume that Ais reduced. Let A be an R-model of ff type for A, and let X = (Spf A) rigdenote the rigid-analytic generic fiber of Spf A. Since A is excellent andsince rigid K-spaces are excellent, [11] Lemma 7.1.9 shows that the space Xis reduced and that we may view A as a subring of Γ(X, O X ) such that thevalue of f in a point x ∈ X agrees with the value of f in the correspondingmaximal ideal of A. Since f(x) = 0 for all x ∈ X, we see that f = 0 as afunction on X and, hence, in A.□2.3. Semi-affinoid spaces.2.3.1. The rigid space associated to a semi-affinoid K-space. Let X = sSp Abe a semi-affinoid K-space. An affine flat formal model of ff type for X isan affine flat formal R-scheme of ff type X together with an identificationof Γ(X, O X ) with an R-model of ff type for A. There is an obvious genericfiber functor urig from the category of affine flat formal R-schemes of ff typeto the category of semi-affinoid K-spaces, given by(Spf A) urig := sSp (A ⊗ R K) .Let X be a flat affine R-model of ff type for X. Berthelot’s constructionyields a rigid K-space X r := X rig together with a K-homomorphismϕ: A → Γ(X r , O X r) ,