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The first Weyl algebra, A, is naturally Z-graded by letting deg x = 1 and deg y = -1. Sue Sierra studied gr-A, the category of graded right A-modules, computing its Picard group and classifying all rings graded equivalent to A. Paul Smith showed that ,in fact, gr-A is equivalent to the category of quasicoherent sheaves on a certain quotient stack. In this dissertation, we generalize results of Sierra and Smith by studying the graded module category of certain generalized Weyl algebras. We show that for a generalized Weyl algebra A(f) with base ring k[z] defined by a quadratic polynomial f, the Picard group of gr-A(f) is isomorphic to the Picard group of gr-A. For each A(f), we also construct a commutative ring whose graded module category is equivalent to the quotient category qgr-A(f), the category gr-A(f) modulo its full subcategory of finite-dimensional modules.

The first Weyl algebra, A, is naturally Z-graded by letting deg x = 1 and deg y = -1. Sue Sierra studied gr-A, the category of graded right A-modules, computing its Picard group and classifying all rings graded equivalent to A. Paul Smith showed that ,in fact, gr-A is equivalent to the category of quasicoherent sheaves on a certain quotient stack. In this dissertation, we generalize results of Sierra and Smith by studying the graded module category of certain generalized Weyl algebras. We show that for a generalized Weyl algebra A(f) with base ring k[z] defined by a quadratic polynomial f, the Picard group of gr-A(f) is isomorphic to the Picard group of gr-A. For each A(f), we also construct a commutative ring whose graded module category is equivalent to the quotient category qgr-A(f), the category gr-A(f) modulo its full subcategory of finite-dimensional modules.