Homological Dimensions of Skew Group Rings

Abstract

Let R be a ring and let H be a subgroup of a finite group G. We consider the weak global dimension, cotorsion dimension and weak Gorenstein global dimension of the skew group ring RσG and its coefficient ring R. Under the assumption that RσG is a separable extension over RσH, it is shown that RσG and RσH share the same homological dimensions. Several known results are then obtained as corollaries. Moreover, we investigate the relationships between the homological dimensions of RσG and the homological dimensions of a commutative ring R, using the trivial RσG-module.

Finiteness properties and homological dimensions of Skew Group rings

“…Since the left 0-perfects rings are the left perfects rings, if we take n = 0 we recover [42,Corollary 2.12]. Taking H as the trivial group we have the following corollary.…”

“…Observe that, if n = 0 we recover [42,Proposition 2.3]. Futhermore, in this case Xiang showed that the cotorsion global dimension of R is less than or equal the cotorsion global dimension of RG [42, Theorem 2.7].…”

“…For commutative R, Gillespie has shown that if R is a Ding-Chen ring, then R [G] is a Ding-Chen ring [23]. Many homological dimensions such as the weak global dimension, cotorsion dimension and weak Gorenstein global dimension of skew group rings have been studied by Xiang [42].…”

Weak dimensions and Gorenstein weak dimensions of group rings

“…In recent decades, representation and homological properties of group rings have been extensively studied (cf. [1], [4], [6], [9], [14], [15], [17], [18], [19]). Among others, Connell in [6] considered necessary and sufficient conditions on R and G so that R[G] have some ring theoretic properties such as being Artinian, regular, self-injective and semiprime.…”