Quantum sections and gauge algebras (Q1802392)

There is an interplay between properties of a filtered ring \(R\) and its associated graded and Rees rings, denoted here \(G(R)\) and \(\widetilde{R}\), respectively. For instance, if \(G(R)\) is a commutative Noetherian domain, one can lift the scheme structure of \(\text{Proj }G(R)\) to \(\text{Proj }\widetilde{R}\) by using quantum sections. The hypothesis on \(G(R)\) can be weakened: For example, one can assume \(G(R)\) to be a non-commutative quadratic Auslander-regular algebra (in this case, \(R\) is called a gauge algebra). The article under review is a survey on these topics, introducing the basic definitions and expanding concrete examples.