Cartan-Eilenberg Gorenstein projective N-complexes

Abstract

We study the notion of Cartan-Eilenberg Gorenstein projective N-complexes. We show that an N-complex G is Cartan-Eilenberg Gorenstein projective if and only if G_n, ${\mathrm{Z}}_n^t(G),{\mathrm{B}}_n^t(G)$ and ${\mathrm{H}}_n^t(G)$ are Gorenstein projective modules for each $n\in \mathbb{Z}$ and $t=1,2,\dots ,N.$ Some applications are induced, for instance, we first establish a relationship between Cartan-Eilenberg Gorenstein projective N-complexes and Gorenstein projective N-complexes. Secondly, we show that an iteration of the procedure used to define the Cartan-Eilenberg Gorenstein projective N-complexes yields exactly the Cartan-Eilenberg Gorenstein projective N-complexes. Finally, we consider Cartan-Eilenberg Gorenstein projectivity of N-complexes along Frobenius extension of rings.

Keywords:

• Gorenstein projective module; Cartan-Eilenberg Gorenstein projective N-complex; Gorenstein projective N-complex; stability; Frobenius extension

2020 Mathematics Subject Classification:

• 18G25
• 18G35
• 18E10

Acknowledgements

The author would like to thank the referee for a careful reading of the paper and for many useful comments and suggestions.

Additional information

Funding

This work was supported by National Natural Science Foundation of China (Grant No.12061061), Fundamental Research Funds for the Central Universities (No.31920190057), XBMUYJRC (No.201406) and Innovation Team Projec of Northwest Minzu University (No.1110130131).