Let $k$ be an algebraically closed field of characteristic $0$.

Let $A := k \langle x,x^{-1},y \rangle /(xy-qyx, x^{d_1}-ay^{d_2})$, where deg$(x)>0$, deg$(y)>0$, $q,a \in k^*$ and $d_1\text{deg}(x)=d_2\text{deg}(y)$. From this, we think of $A$ as a $\mathbb{Z}$-graded algebra over $k$.

We also assume $q^{d_1}=q^{d_2}=1$.

Let $\text{Mod}(A)$ be the category of right $A$-modules and $\text{Gr}(A)$ be the category of graded right $A$-modules.

We define the graded global dimension $\text{gr.gl.dim}(A) := \text{sup}_{M \in \text{Gr}(A)} \text{ gr.pd}(M)$ and the global dimension $\text{gl.dim}(A):= \text{sup}_{M \in \text{Mod(A)}} \text{ pd}(M)$, where $\text{ gr.pd}(M)$ is the minimum length of graded projective resolution of $M$ and $\text{pd}(M)$ is the minimum length of projective resolutions of $M$.

**Question**

$\text{gr.gl.dim}(A) = 0$ ?

It is known that $\text{gr.gl.dim}(A) \leq \text{gl.dim}(A)$ and $\text{gr.pd}(M) = \text{pd}(M)$ for any $M \in \text{Gr}(A)$.

Thank you.