Rocky Mountain Journal of Mathematics

$\mathcal {W}$-Gorenstein $N$-complexes

Bo Lu, Jiaqun Wei, and Zhenxing Di

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Given an integer $N\geq 2$ and a self-orthogonal subcategory $\mathcal {W}$ of an abelian category $\mathscr {A}$, we investigate the $\mathcal {W}$-Gorenstein $N$-complexes. We show that an $N$-complex $G$ is $\mathcal {W}$-Gorenstein if and only if $G$ is an $N$-complex consisting of $\mathcal {W}$-Gorenstein objects in $\mathscr {A}$. As an application, we improve a result of Estrada.

Article information

Rocky Mountain J. Math., Volume 49, Number 6 (2019), 1973-1992.

First available in Project Euclid: 3 November 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 18G25: Relative homological algebra, projective classes
Secondary: 18G35: Chain complexes [See also 18E30, 55U15] 18E10: Exact categories, abelian categories

$\mathcal {W}$ $N$-complex $\mathcal {W}$-Gorenstein $N$-complex $\mathcal {W}$-Gorenstein object Gorenstein injective $N$-complex.


Lu, Bo; Wei, Jiaqun; Di, Zhenxing. $\mathcal {W}$-Gorenstein $N$-complexes. Rocky Mountain J. Math. 49 (2019), no. 6, 1973--1992. doi:10.1216/RMJ-2019-49-6-1973.

Export citation


  • M. Auslander and M. Bridger, Module theory, Mem. Amer. Math. Soc. 94, Amer. Math. Soc., Providence, RI (1969).
  • H. Cartan and S. Eilenberg, Homological algebra, Princeton University Press (1956).
  • L.W. Christensen, Semi-dualizing complexes and their Auslander categories, Trans. Amer. Math. Soc. 353 (2001), 1839–1883.
  • M. Dubois-Violette and R. Kerner, Universal $q$-differential calculus and $q$-analog of homological algebra, Acta Math. Univ. Comenian. (N.S.) 65 (1996), 175–188.
  • E.E. Enochs, S. Estrada and J.R. García Rozas, Gorenstein categories and Tate cohomology on projective schemes, Math. Nachr. 281 (2008), 525–540.
  • E.E. Enochs and J.R. García Rozas, Gorenstein injective and projective complexes, Comm. Algebra 26 (1998), 1657–1674.
  • E.E. Enochs and O.M.G. Jenda, Gorenstein injective and Gorenstein projective modules, Math. Z. 220 (1995), 611–633.
  • E.E. Enochs and O.M.G. Jenda, Relative homological algebra, De Gruyter Expositions in Math. 30, Walter de Gruyter & Co., Berlin (2000).
  • E.E. Enochs, O.M.G. Jenda and J.A. López-Ramos, Covers and envelopes by $V$-Gorenstein modules, Comm. Algebra 33 (2005), 4705–4717.
  • S. Estrada, Monomial algebras over infinite quivers: applications to $N$-complexes of modules, Comm. Algebra 35 (2007), 3214–3225.
  • H.B. Foxby, Gorenstein modules and related modules, Math. Scand. 31 (1972), 267–284.
  • Y.X. Geng and N.Q. Ding, $\mathcal{W}$-Gorenstein modules, J. Algebra 325 (2011), 132–146.
  • J. Gillespie and M. Hovey, Gorenstein model structures and generalized derived categories, Proc. Edinb. Math. Soc. 53 (2010), 675–696.
  • E.S. Golod, $G$-dimension and generalized perfect ideals, Trudy. Mat. Inst. Steklov. 165 (1984), 62–66.
  • H. Holm and P. Jørgensen, Semi-dualizing modules and related Gorenstein homological dimensions, J. Pure Appl. Algebra 205 (2006), 423–445.
  • H. Holm and D. White, Foxby equivalence over associative rings, J. Math. Kyoto Univ. 47 (2007), 781–808.
  • O. Iyama, K. Kato and J.I. Miyachi, Derived categories of $N$-complexes, J. Lond. Math. Soc. 96 (2017), 687–716.
  • M.M. Kapranov, On the $q$-analog of homological algebra, preprint (1996)..
  • L. Liang, N.Q. Ding and G. Yang, Some remarks on projective generators and injective cogenerators, Acta Math. Sin. (Engl. Ser.) 30 (2014), 2063–2078.
  • Z.K. Liu and C.X. Zhang, Gorenstein injective complexes of modules over Noetherian rings, J. Algebra 321 (2009), 1546–1554.
  • W. Mayer, A new homology theory, I, Annals Math. 43 (1942), 370–380.
  • W. Mayer, A new homology theory, II, Annals Math. 43 (1942), 594–605.
  • S. Sather-Wagstaff, T. Sharif and D. White, Stability of Gorenstein categories, J. Lond. Math. Soc. 77 (2008), 481–502.
  • W.V. Vasconcelos, Divisor theory in module categories, North-Holland Mathematics Studies 14, Notas de Matemática 53, North-Holland Publishing Co., Amsterdam (1974).
  • D. White, Gorenstein projective dimension with respect to a semidualizing module, J. Commut. Algebra 2 (2010), 111–137.
  • X.Y. Yang and N.Q. Ding, The homotopy category and derived category of $N$-complexes, J. Algebra 426 (2015), 430–476.
  • X.Y. Yang and Z.K. Liu, Gorenstein projective, injective, and flat complexes, Comm. Algebra 39 (2011), 1705–1721.
  • C.H. Yang and Jiayou L. Liang, Gorenstein injective and projective complexes with respect to a semidualizing module, Comm. Algebra 40 (2012), 3352–3364.