Ding projective complexes with respect to a semidualizing module

TL;DR

This paper introduces the concept of DC-projective complexes relative to a semidualizing module over a commutative ring, characterizes their properties, stability, and defines their dimension.

Contribution

It defines DC-projective complexes, provides characterizations, studies their stability under iteration, and introduces the notion of DC-projective dimension.

Findings

DC-projective complexes are characterized by their degree modules and Hom conditions.

The stability of DC-projective complexes under iterative procedures is established.

A new notion of DC-projective dimension for complexes is introduced.

Abstract

Let R be a commutative ring and C a semidualizing R-module. In this article, we introduce and investigate the notion of DC-projective complexes. We first prove that a complex X is DC-projective if and only if each degree of X is a DC-projective module and Hom(X;H) is exact for any C-flat complex H. As immediate consequences of this result, some properties of DC-projective complexes are given. Secondly, we investigate a kind of stability of DC-projective complexes by showing that an iteration of the procedure used to define the DC-projective complexes yields exactly the DC-projective complexes. Finally, We introduce and characterize the notion of DC-projective dimension of complexes.