Keep the setting of my previous question and let $I := k[x_1, \dots, x_n] \cdot A_{>0}$ be an ideal of the algebra $k[x_1, \dots, x_n]$ generated by the set $A_{>0}$. It is clear that $I$ is a graded subspace of $k[x_1, \dots, x_n]$ and one has a vector space decomposition $k[x_1, \dots, x_n]/I = \bigoplus_{i \ge 0} (k^i[x_1, \dots, x_n]/I_i)$. Let $H \subset k[x_1, \dots, x_n]$ be a graded vector subspace such that $k[x_1, \dots, x_n] = I \oplus H$. What is the easiest way to see that the following three properties are equivalent? (Or can provide a reference to where this is proven...)
- $k[x_1, \dots, x_n]$ is free as an $A$-module.
- The map $A \otimes_k H \to k[x_1, \dots, x_n]$ induced by multiplication in the algebra $k[x_1, \dots, x_n]$ is a vector space isomorphism.
- One has an equality: $P_{k[x_1, \dots, x_n]}(t) = P_A(t) \cdot P_{k[x_1, \dots, x_n]/I}(t)$, of formal power series.