Exercise 1, § 7

Let $\Gamma \subset \mathbf{N}$ be a submonoid of $\mathbf{N}$ such that $\mathbf{N} - \Gamma$ is a finite set. Let $(a_1, ..., a_r)$ $(0 < a_1 < ... < a_r)$ be a minimal system of generators of $\Gamma$, and let $k$ be a field. Put $A = k[\Gamma]$. a) Prove that $A$ is isomorphic to the sub-$k$-algebra of $k[T]$ generated by $T^{a_1}, ..., T^{a_r}$. b) Let $m$ be the maximal ideal of $A$ generated by $T^{a_1}, ..., T^{a_r}$; let $A_m$ be the local ring of $A$ at $m$, and put $q = mA_m$. Prove that one has $\dim_k q/q^2 = r$ and $e_q(A_m) = a_1$.