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.

Key words： Cartan-Eilenberg \mathcal{W}N-complex; Cartan-Eilenberg projective N- complex; Cartan-Eilenberg injective N-complex; Cartan-Eilenberg balance

References

1 Beligiannis A. Relative homological algebra and purity in triangulated categories. J Algebra, 2000, 227(1): 268–361

2 Cartan H, Eilenberg S. Homological Algebra. Princeton: Princeton Univ Press, 1956

3 Cibils C, Solotar A, Wisbauer R. N-Complexes as functors, amplitude cohomology and Fusion rules. Comm Math Phys, 2007, 272(3): 837–849

4 Dubois-Violette M, Kerner R. Universal q-differential calculus and q-analog of homological algebra. Acta Math Univ Comenian (N S), 1996, 65(2): 175–188

5 Enochs E E. Cartan-Eilenberg complexes and resolutions. J Algebra, 2011, 342: 16–39

6 Estrada S. Monomial algebras over infinite quivers, applications to N-complexes of modules. Comm Algebra, 2007, 35 (10): 3214–3225

7 Geng Y X, Ding N Q. W -Gorenstein modules. J Algebra, 2011, 325: 132–146

8 Gillespie J. The homotopy category of N-complexes is a homotopy category. J Homotopy Relat Struct, 2015, 10: 93–106

9 Gillespie J, Hovey M. Gorenstein model structures and generalized derived categories. Proc Edinb Math Soc, 2010, 53(3): 675–696

10 Iyama O, Kato K, Miyachi J I. Derived categories of N-complexes. J Lond Math Soc, 2017, 96(3): 687–716

11 Kapranov M M. On the q-analog of homological algebra. arXiv: q-alg/9611005

12 Lu B, Di Z X. Gorenstein cohomology of N-complexes. J Algebra Appl, 2019, https://doi.org/10.1142/S0219498820501741

13 Lu B, Liu Z K. Cartan-Eilenberg complexes with respect to cotorsion pairs. Arch Math (Basel), 2014, 102(1): 35–48

14 Lu B, Liu Z K. Cartan-Eilenberg FP-injective complexes. J Aust Math Soc, 2017, 103: 387–401

15 Lu B, Ren W, Liu Z K. A note on Cartan-Eilenberg Gorenstein categories. Kodai Math J, 2015, 38: 209–227

16 Lu B, Wei J Q, Di Z X. W -Gorenstein N-complexes. Rocky Mountain J Math, 2019, 49(6): 1971–1992

17 Mayer W. A new homology theory I. Ann of Math, 1942, 43: 370–380

18 Mayer W. A new homology theory II. Ann of Math, 1942, 43: 594–605

19 Verdier J L. Des catégories dérivées des catégories abéliennes. Astérisque, Vol 239. Paris: Soc Math France, 1997

20 White D. Gorenstein projective dimension with respect to a semidualizing module. J Commut Algebra, 2010, 2: 111–137

21 Yang G, Liang L. Cartan-Eilenberg Gorenstein projective complexes. J Algebra Appl, 2014, 13(1): 1350068

22 Yang G, Liang L. Cartan-Eilenberg Gorenstein at complexes. Math Scand, 2014, 114: 5–25

23 Yang X Y, Cao T Y. Cotorsion pairs in CN(d). Algebra Colloq, 2017, 24: 577–602

24 Yang X Y, Ding N Q. The homotopy category and derived category of N-complexes. J Algebra, 2015, 426: 430{476

25 Yang X Y, Wang J P. The existence of homotopy resolutions of N-complexes. Homology Homotopy Appl, 2015, 17: 291–316