the first weyl algebra $a = k\langle x, y \rangle/(xy - yx - 1)$ is $\mathbb{z}$-graded with deg x = 1 and deg y = -1. susan sierra and paul smith studied the category of graded modules over a, showing that this category was equivalent to coherent sheaves on a certain quotient stack. in this talk, we investigate the graded module categories over $\mathbb{z}$-graded rings called generalized weyl algebras. we construct commutative rings with equivalent graded module categories. in the pre-talk, we will discuss some preliminaries on categories and graded rings before giving an overview of noncommutative projective geometry. please note: there will be a pre-talk for graduate 2022年亚洲世界杯预选赛 from 2:30 - 3:00. the regular talk will begin at 3:00.