A smooth non-compactly generated category

Let $A=\mathbb{Q}[t^{\frac{1}{2^n}}:n\ge 0]$ and consider $A[u,u^{-1};\sigma ]$, a skew Laurent polynomial ring, where $\sigma (t^q)=t^{2q}$. We study the kernel of derived extension of scalars along $A[u,u^{-1};\sigma ]⟶\mathbb{Q}[u,u^{-1}],$ the quotient by $\mathfrak{m}A[u,u^{-1};\sigma ]$, with $\mathfrak{m}=(t^{\frac{1}{2^n}}:n\ge 0)$. A finite free resolution of $\mathfrak{m}A[u,u^{-1};\sigma ]$ over $A[u,u^{-1};\sigma ]{\otimes }_{\mathbb{Q}}(A[u,u^{-1};\sigma ]{)}^{\mathrm{o}\mathrm{p}}$ proves smoothness. A specialization inequality for matrix ranks supplies a nonzero exact colimit-preserving functor that vanishes on every compact object, proving failure of compact generation.