Exercise 4, § 5

Let $A$ be a local noetherian regular ring. Then one has $\dim(A) = \mathrm{dh}(A)$. (To show that one has $\mathrm{dh}(A) \leq \dim(A)$, use the Koszul complex associated with a system of coordinates. Determine then the groups $\mathrm{Ext}_A^i(k_A, k_A)$.)