Let $R$ be a quotient of the polynomial ring $\mathbb{C}[x_1,\dots , x_n]$. We fix a $\mathbb{C}^*$ action on $R$ which preserve homogenous components and the multiplication. (The geometric analogue of this is a $\mathbb{C}^*$ action on Proj($R$).)
We consider the induced $\mathbb{C}^*$ action on $R[t]$; i.e. $\mathbb{C}^*$ acts trivially on $t$. This gives a grading on $R[t]$, and we will consider the leading order term with respect to this grading.
Let $\chi$ be a finitely generated sub-algebra of $R[t]$ which is not preserved by $\mathbb{C}^*$. We let $\bar{\chi}$ to be the algebra generated by the leading order terms of the elements of $\chi$.
I would like to know if $\bar{\chi}$ is finitely generated. More specifically, there should be a finite set of generators of $\chi$ - a Grobner basis - such that their leading order terms generate $\bar{\chi}$.
Thanks