Exercise 2, § 2

Let V be a discrete valuation ring, with field of fractions K and residue field k. Denote by A the ring K × k and by ρ : V → A the homomorphism deduced from the canonical homomorphisms of V into K and k. The homomorphism ρ is injective, the mapping “ρ : Spec(A) → Spec(V) is surjective, A is a finitely generated V-algebra and one has dim(A) < dim(V).