2022/02/22 by Li, Haonan, Wu, Quanshui · 1 citation
#FOS: Mathematics #Quantum Algebra (math.QA) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2202.10735
We define generalized Koszul modules and rings and develop a generalized Koszul theory for ℕ-graded rings with the degree zero part noetherian semiperfect. This theory specializes to the classical Koszul theory for graded rings with degree zero part artinian semisimple developed by Beilinson-Ginzburg-Soergel and the ungraded Koszul theory for noetherian semiperfect rings developed by Green and Martinéz-Villa. Let A be a left finite ℕ-graded ring generated in degree 1 with A0 noetherian semiperfect, J be its graded Jacobson radical and S=A/J. By the Koszul dual of A we mean the Yoneda Ext ring \underlineExtA^\bullet(S,S). If A is a generalized Koszul ring and M is a generalized Koszul module, then it is proved that the Koszul dual of the Koszul dual of A is GrJ A and the Koszul dual of the Koszul dual of M is GrJ M. If A is a locally finite algebra, then the following statements are proved to be equivalent: A is generalized Koszul; the Koszul dual \underlineExtA^\bullet(S,S) of A is (classically) Koszul; GrJ A is (classically) Koszul; the opposite ring Aop of A is generalized Koszul. It is also proved that if A is generalized Koszul with finite global dimension then A is generalized AS regular if and only if the Koszul dual of A is self-injective.