Exercise 11, § 1

Let $(A_i, f_{ij})$ be an inductive system of rings relative to a filtered ordered set I, with inductive limit A. Now prove that one has \dim(A) \leq \liminf \dim(A_i) and that one has equality when the homomorphisms f_{ij} are faithfully flat. Give an example where the homomorphisms f_{ij} are flat and where one has \dim(A) < \liminf \dim(A_i) (take I = \mathbf{N}, A_n = \mathbf{Z}[n^{-1}], A = \mathbf{Q}).