Exercise 10, § 5

Let $A$ be a valuation ring whose group of orders $\Gamma$ possesses the property (GOD). a) Prove that $\mathrm{dh}(A) \leq 2$. (One will note that every ideal of $A$ is a countable inductive limit of principal ideals and one will use the results of Exercises 7 and 8.) b) Every valuation ring of Krull dimension 1 is of homological dimension $\leq 2$. It is of homological dimension 2 if and only if it is not noetherian. (Use Exercise 9, a) and A, X, p. 208, Exercise 12.) c) For every integer $n \geq 1$, there exists a ring of Krull dimension equal to $n$ and of homological dimension equal to 2. (Use Exercise 9, b) and A, X, p. 208, Exercise 12.)