Exercise 4, § 3

Let k be a field, n an integer ≥ 1. For each integer r ∈ {0, n}, let M_r be the vector subspace of k(X_1, ..., X_n) generated by the monomials X_1^{\alpha_1} ... X_r^{\alpha_r} with α_1, ..., α_r in Z and α_r > 0. Put a_r = M_r + ... + M_n, for 0 ≤ r ≤ n + 1. Then a_0 is a subring of k(X_1, ..., X_n), and the a_r are prime ideals of a_0. Put A = (a_0)_{a_1} and p_r = (a_r)_{a_1} for 1 ≤ r ≤ n + 1. Prove that the only prime ideals of the local ring A are the p_r, that one has ht(p_r) = n + 1 − r and dim(A) = n. Moreover p_1 is principal.