# Iff criterion for augmentation ideal to be finitely generated.

Let $A \subset k[x_1, \dots, x_n]$ be a subalgebra, which is also a graded subspace$$A = \bigoplus_{i \ge 0} A_i.$$One can write$$A = A_0 \oplus A_{> 0},$$where we have$$A_0 = k^0[x_1, \dots, x_n] = k$$and$$A_{> 0} := \bigoplus_{i \ge 1} A_i$$is a graded ideal of $A$, called the augmentation ideal.

Can anyone provide me a reference of the fact (or supply a proof) that the ideal $A_{> 0}$ is finitely generated (as an $A$-module) if and only if $A$ is finitely generated as a $k$-algebra?

• May I ask whether you know anything about the grading? Is the grading of $f(x_1,\cdots, x_n)\in A$ the maximal degree of monomials in $f$? – Ying Zhou Sep 16 '15 at 1:12
• This question has been answered (in a comment) at MathOverflow mathoverflow.net/questions/214748/… – Christopher Drupieski Sep 16 '15 at 14:46