The noncommutative schemes of generalized Weyl algebras

The first Weyl algebra over k, A1=k〈x,y〉/(xy−yx−1) admits a natural Z-grading by letting deg⁡x=1 and deg⁡y=−1. Smith showed that gr­A1 is equivalent to the category of quasicoherent sheaves on a certain quotient stack. Using autoequivalences of gr­A1, Smith constructed a commutative ring C, graded by finite subsets of the integers. He then showed gr­A1≡gr­(C,Zfin). In this paper, we prove analogues of Smith's results 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.