Let $R$ be a noncommutative finitely generated $\mathbb{C}$-algebra such that its center $S$ is smooth (in commutative sense) and $R$ is finite over $S$. Is there Castelnuovo-Mumford regularity theorem for this kind of $\mathbb{C}$-algebra $R$? I have $$ R=\mathbb{C}\langle x,y,z \rangle/(xy=\zeta_1yx,yz=\zeta_2zy,zx=\zeta_3xz,x^3+y^3+z^3), $$ where $\zeta_i$ is a 3rd root of unity. I am aware of CM regularity theorem for AS regular algebra but not sure if it works for this example. Thank you in advance.
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