Cartan-Eilenberg N-complexes with respect to self-orthogonal subcategories

Abstract

Let R be an arbitrary associated ring. For an integer N≥2 and a self-orthogonal subcategory \mathcal{W} of R-modules, we study the notion of Cartan-Eilenberg \mathcal{W}N-complexes. We show that an N-complex X is Cartan-Eilenberg \mathcal{W} if and only if X\cong X\text{'}\oplus X\text{'}\text{'} in which X\text{'} is a \mathcal{W}N-complex and X\text{'}\text{'} is a graded R-module with {{\displaystyle X}}_n^{\text{'}\text{'}}\in \mathcal{W} for all n\in \mathbb{Z}. As applications of the result, we obtain some characterizations of Cartan-Eilenberg projective and injective N-complexes, establish Cartan and Eilenberg balance of N-complexes, and give some examples for some fixed integers N to illustrate our main results.

Keywords

Cartan-Eilenberg \mathcal{W}N-complex / Cartan-Eilenberg projective N- complex / Cartan-Eilenberg injective N-complex / Cartan-Eilenberg balance

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