Exercise 15, § 7

Let k be a perfect field of characteristic p > 0, $(T_n)_{n \in \mathbf{N}}$ a family of indeterminates, and K the field $k((T_n)_{n \in \mathbf{N}})$. Let A be the subring of the ring of formal series K[[X, Y]] generated by K^p[[X, Y]] and K. The ring A is local noetherian regular of dimension 2 and its completion is K[[X, Y]] (§ 6, exerc. 8). For every integer n, denote by $p_n$ the element X − T_nY of A, by $q_n$ the product $p_0 \cdots p_n$, and put $c = \sum_{n \in \mathbf{N}} q_n T_n$. Denote by B the sub-A-algebra of K[[X, Y]] generated by c.

a) Let n be an integer > 0 and p a prime ideal of B of height 1 which contains $p_n$. Now prove that the ring B_p is not normal (for every integer m, one will put $c_m = (c - \sum_{i=0}^{m-1} q_i T_i)/q_m$ and one will show that $B_p = A_{Ap_n}[c_{n-1}]$, so that $c_n$ is not in B_p , whereas $c_n^p$ belongs to A).

b) Deduce that the set of prime ideals p of B such that the ring B_p is regular contains no nonempty open subset of Spec(B) (one will note that B is a faithfully flat A-module).