Simple \(\mathbb{Z} \)-graded domains of Gelfand-Kirillov dimension two (Q2197569)

The authors prove that if \(A\) is a simple finitely generated \(\mathbb{Z}\)-graded domain of Gelfand-Kirillov dimension 2 such that all homogeneous components of \(A\) are non-zero, then there exist an abelian group \(\Gamma\) and a commutative \(\Gamma\)-graded ring \(B\) such that the category of graded right \(A\)-modules is equivalent to the category of \(\Gamma\)-graded \(B\)-modules. As a consequence, the category of \(\mathbb{Z}\)-graded right \(A\)-modules is equivalent to the category of quasicoherent sheaves on a certain quotient stack.