Exercise 9, § 9

Let $\Gamma$ be a submonoid of the additive monoid $\mathbf{N}$, such that $\mathbf{N} - \Gamma$ is finite; denote by $c$ the smallest integer such that $\Gamma$ contains all the integers $\geq c$. Let $k$ be a field, $B$ the ring $k[[T]]$, $K = k((\Gamma))$ its field of fractions, and $A$ the sub-$k$-algebra of $B$ generated by the elements $T^\gamma$ for $\gamma \in \Gamma$.

a) Prove that $A$ is an integral noetherian local ring of dimension 1, whose field of fractions is $K$ and whose integral closure is $B$; the ideal $c = A : B$ is equal to $BT^c$.

b) For $A$ to be a Gorenstein ring, it is necessary and sufficient that one has $c = 2 \mathrm{Card}(\mathbf{N} - \Gamma)$.

c) Let $\Omega$ be the sub-$A$-module of $K$ generated by the elements $T^{-\alpha}$ for $\alpha \notin \Gamma$. Prove that $K/\Omega$ is a Matlis $A$-module (use prop. 2 of § 8), and that $\Omega$ is a dualizing $A$-module.