The extension dimension of group graded rings

TL;DR

The paper introduces the graded extension dimension $gr.ext.dim(R)$ for group graded rings and proves that for strongly graded rings one has $ext.dim(R)=gr.ext.dim(R)=ext.dim(R_e)$. It establishes that graded (strong) Morita and separable equivalences preserve extension dimensions under suitable conditions, and it provides concrete applications to Morita context rings and multiplier rings to illustrate invariance results in graded settings. The work connects graded homological complexity with ungraded counterparts, showing that graded structures retain the essential extension-theoretic properties when the grading is well-behaved (e.g., strong grading) and the group is finite where necessary. Overall, the results offer a robust framework for transferring extension-dimension and finite representation-type information across graded and ungraded contexts, with practical implications for constructing and comparing graded algebras via common equivalences and decompositions.

Abstract

In this paper, we introduce the concept of graded extension dimension for a group graded ring R, denoted by gr.ext.dim(R). We prove that when R is strongly graded, its graded extension dimension coincides with the non-graded extension dimension of both R itself and its degree-zero subring Re. Furthermore, we demonstrate that graded equivalence and graded separable equivalence preserve the extension dimension under appropriate conditions.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Remark 2.2
• Lemma 2.3
• Corollary 2.4
• Remark 2.5
• Definition 3.1
• Remark 3.2
• Lemma 3.3