The noncommutative schemes of generalized Weyl algebras

The first Weyl algebra over $k$, $A_1=k⟨x,y⟩\mathrm{/}(xy-yx-1)$ admits a natural $\mathbb{Z}$-grading by letting $\mathrm{deg}x=1$ and $\mathrm{deg}y=-1$. Paul Smith showed that $\mathrm{gr}-A_1$ is equivalent to the category of quasicoherent sheaves on a certain quotient stack. Using autoequivalences of $\mathrm{gr}-A_1$, Smith constructed a commutative ring $C$, graded by finite subsets of the integers. He then showed $\operatorname{gr}- A_1 \equiv \operatorname{gr}- (C, $\mathbb{Z}$_{$\mathrm{f}\mathrm{i}\mathrm{n}$})$. In this paper, we generalize results of Smith by using autoequivalences of a graded module category to construct rings with equivalent graded module categories. For certain generalized Weyl algebras, we use autoequivalences defined in a companion paper so that these constructions yield commutative rings.