Injective generation for graded rings
Panagiotis Kostas, Chrysostomos Psaroudakis
TL;DR
The paper develops a comprehensive framework for injective generation in graded rings, centering on graded injective generation and its connections to Morita context rings, covering rings, and cleft/twisted extensions. It proves graded injectives generate for every $oldsymbol{ ext{$oldsymbol{ ext{ extepsilon}}$-commutative}}$ graded Noetherian ring, and establishes reduction techniques that transfer injective-generation properties between graded and ungraded settings, Morita-context constructions, and tensor/twisted-tensor products. Key contributions include precise criteria under which injective generation passes from a base ring to tensor rings with perfect nilpotent bimodules, a cleft-extension theory that reduces injective generation to $ heta$-extensions, and an extension to twisted tensor products. These results provide new tools for verifying the finitistic dimension conjecture in broad classes of algebras and illuminate the interplay between grading, Morita theory, and homological methods. The work also identifies open problems and directions for extending these transfer principles to more general gradings and constructions.
Abstract
In this paper we investigate injective generation for graded rings. We first examine the relation between injective generation and graded injective generation for graded rings. We then reduce the study of injective generation for graded rings to the study of injective generation for certain Morita context rings and we provide sufficient conditions for injective generation of the latter. We then provide necessary and sufficient conditions so that injectives generate for tensor rings and for trivial extension rings. We provide two proofs for the class of tensor rings, the one uses covering theory and the other uses the framework of cleft extensions of module categories. We finally prove injective generation for twisted tensor products of finite dimensional algebras.