2015/08/24 by M. Rahmani, Rahmani, M., A. J. Taherizadeh +1
Mathematics · #13C05 #13C11 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1508.05813
openalex publication_date 2015/08/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let R be a Noetherian ring and let C be a semidualizing R-module. In this paper, by using relative homological dimensions with respect to C, we impose various conditions on C to be dualizing. First, we show that C is dualizing if and only if there exists a Cohen-Macaulay R-module of type 1 and of finite G C -dimension. This result extends Takahashi \cite[Theorem 2.3]T as well as Christensen \cite[Proposition 8.4]C. Next, as a generalization of Xu \cite[Theorem 3.2]X2, we show that C is dualizing if and only if for an R-module M, the necessary and sufficient condition for M to be C-injective is that πi(\fp , M) = 0 for all \fp ∈ \Spec(R) and all i ≠ \h(\fp) , where πi is the invariant dual to the Bass numbers defined by E.Enochs and J.Xu \citeEX. We use the later result to give an explicit structure of the minimal flat resolution of \H\fmd(R) , where (R, \fm) is a d -dimensional Cohen-Macaulay local ring possessing a canonical module. As an application, we compute the torsion product of these local cohomology modules.