Exercise 15, § 8

Let $\sigma$ be an automorphism of the ring A. a) We denote by $A_\sigma[X]$ the ring defined in VIII, § 1, No. 3. Now prove that

(The inequality $\mathrm{dh}(A_\sigma[X]) \geq \mathrm{dh}(A) + 1$ follows from Exercise 14, b); for the opposite inequality, take inspiration from the proof of X, p. 142, Lemma 3, a).) b) We denote by $A_\sigma[[X]]$ the ring defined in IV, § 4, Exercise 8. Now prove that if A is noetherian, one has $\mathrm{dh}(A_\sigma[[X]]) = \mathrm{dh}(A) + 1$ (use Exercise 14).