Exercise 2, § 7

Let $k$ be a field, $m$ an integer $\geq 1$, and $I$ an ideal of the polynomial ring $k[T_1, ..., T_m]$ generated by monomials represented by points $M_1, ..., M_s$ of the “positive quadrant” $\mathbf{R}_+^m$ of $\mathbf{R}^m$. Let $\mathscr{N}$ denote the convex hull in $\mathbf{R}_+^m$ of the union of the sets $M_i + \mathbf{R}_+^m$ for $1 \leq i \leq s$. a) Put $m = (T_1, ..., T_m)$, $A = k[T_1, ..., T_m]_m$ and $q = I.A$. Prove that $A/q$ is of finite length if and only if the volume of $\mathbf{R}_+^m - \mathscr{N}$ is finite. b) Prove the equality $e_q(A) = m! \ \mathrm{Vol}(\mathbf{R}_+^m - \mathscr{N})$.