The noncommutative schemes of generalized Weyl algebras

TL;DR

This paper generalizes Smith's work on the Weyl algebra by using autoequivalences to construct rings with equivalent graded module categories, including commutative rings for certain generalized Weyl algebras.

Contribution

It extends Smith's framework to a broader class of generalized Weyl algebras using autoequivalences, producing new rings with equivalent graded module categories.

Findings

Constructed rings with equivalent graded module categories for generalized Weyl algebras.

Used autoequivalences to obtain commutative rings in specific cases.

Generalized Smith's results to a wider algebraic context.

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.