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arXiv · 1606.07800

The noncommutative schemes of generalized Weyl algebras

Robert Won
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Abstract

The first Weyl algebra over $k$, $A_1 = k \langle x, y\rangle/(xy-yx - 1)$ admits a natural $\mathbb{Z}$-grading by letting $\operatorname{deg} x = 1$ and $\operatorname{deg} y = -1$. Paul Smith showed that $\operatorname{gr}- A_1$ is equivalent to the category of quasicoherent sheaves on a certain quotient stack. Using autoequivalences of $\operatorname{gr}- A_1$, Smith constructed a commutative ring $C$, graded by finite subsets of the integers. He then showed $\operatorname{gr}- A_1 \equiv \operatorname{gr}- (C, \mathbb{Z}_{\mathrm{fin}})$. 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.

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Robert Won. 2016-06-24. The noncommutative schemes of generalized Weyl algebras. https://doi.org/10.1016/j.jalgebra.2018.04.004

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