LAUSR

Abstract The first Weyl algebra over k , A 1 = k 〈 x , y 〉 / ( x y − y x − 1 ) admits a natural Z -grading by letting deg ⁡ x = 1 and deg ⁡ y = − 1 . Smith showed that gr ­ A 1 is equivalent to the category of quasicoherent sheaves on a certain quotient stack. Using autoequivalences of gr ­ A 1 , Smith constructed a commutative ring C, graded by finite subsets of the integers. He then showed gr ­ A 1 ≡ gr ­ ( C , Z fin ) . 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.