A smooth non-compactly generated category

Let $A=\mathbb{Q}[t^{\frac{1}{2^n}}:n\geq 0]$ and consider $A[u,u^{-1};σ]$, a skew Laurent polynomial ring, where $σ(t^q)=t^{2q}$. We study the kernel of derived extension of scalars along $ A[u,u^{-1};σ]\longrightarrow\mathbb{Q}[u,u^{-1}], $ the quotient by $\mathfrak{m}A[u,u^{-1};σ]$, with $\mat