Exercise 11, § 2

Let $B$ be a noetherian integrally closed ring, $G$ a finite group of automorphisms of $B$, $A$ the ring of invariants of $G$.

a) Suppose that for every prime ideal $p$ of $A$ of height 1 and every prime ideal $q$ of $B$ lying above $p$, the valuation $v_q$ is unramified with respect to $v_p$ (VI, § 8, No. 1). Let $i : C(A) \to C(B)$ be the canonical homomorphism on the groups of divisor classes (VII, § 1, No. 10); prove that the kernel of $i$ is isomorphic to the cohomology group $H^1(G, B^*)$ (A, X, p. 111; let $L$ be the field of fractions of $B$; consider the exact sequence of $\mathbf{Z}^{(G)}$-modules

by observing that $H^0(G, D(B))$ identifies with $D(A)$ and that $H^1(G, L^*)$ is zero).

b) Let $k$ be a field of characteristic 2; take for $B$ the ring $k[X_1, \ldots, X_4]$, and for $G$ the group of automorphisms of the $k$-algebra $B$ generated by the automorphism $\sigma$ such that $\sigma(X_i) = X_{i+1}$ for $1 \leq i \leq 3$ and $\sigma(X_4) = X_1$. Deduce from a) that the ring $A$ of invariants of $G$ is factorial.

c) Let $m$ be the trace on $A$ of the ideal $(X_1, \ldots, X_4)$; put $s = X_1 + \ldots + X_4$, $t = (X_1 + X_3)(X_2 + X_4)$, $u = (X_1 + X_2)(X_3 + X_4)(X_1 + X_4)(X_2 + X_3)$, $v = X_1 X_2 X_3 X_4$. Prove that the sequence $(s, t, u, v)$ is transversal but not completely transversal for $A_m$. (Let $x = X_1^2(X_2 + X_3) + X_2^2(X_3 + X_4) + X_3^2(X_4 + X_1) + X_4^2(X_1 + X_2)$, $y = X_1 X_3 + X_2 X_4$, $w = X_1 X_2 X_3 + X_2 X_3 X_4 + X_3 X_4 X_1 + X_4 X_1 X_2$; prove that one has $sw + y^2 = u$ and that $xy + tw$ is divisible by $s$, so that $ux$ belongs to the ideal $(s, t)$.)

d) Prove that $A$ is not a Macaulay ring, and that the sub-$A$-module $A.1_B$ is not a direct factor in $B$.