Czechoslovak Mathematical Journal, online first, 18 pp.

(Strongly) Gorenstein injective modules over upper triangular matrix Artin algebras

Chao Wang, Xiaoyan Yang

Received July 1, 2016.  First published March 1, 2017.

Abstract:  Let $\Lambda=\left(\begin{smallmatrix} A&M 0&B \end{smallmatrix}\right)$ be an Artin algebra. In view of the characterization of finitely generated Gorenstein injective $\Lambda$-modules under the condition that $M$ is a cocompatible $(A,B)$-bimodule, we establish a recollement of the stable category $\overline{\rm Ginj(\Lambda)}$. We also determine all strongly complete injective resolutions and all strongly Gorenstein injective modules over $\Lambda$.
Keywords:  (strongly) Gorenstein injective module; upper triangular matrix Artin algebra; triangulated category; recollement
Classification MSC:  18G25, 16E65, 18E30
DOI:  10.21136/CMJ.2017.0346-16

PDF available at:  Springer   Myris Trade   Institute of Mathematics CAS

[1] F. W. Anderson, K. R. Fuller: Rings and Categories of Modules. Graduate Texts in Mathematics 13, Springer, New York (1992). DOI 10.1007/978-1-4612-4418-9 | MR 1245487 | Zbl 0765.16001
  [2] M. Auslander, I. Reiten, S. O. Smalo: Representation Theory of Artin Algebras. Cambridge Studies in Advanced Mathematics 36, Cambridge University Press, Cambridge (1995). DOI 10.1017/CBO9780511623608 | MR 1314422 | Zbl 0834.16001
  [3] A. Beligiannis: On algebras of finite Cohen-Macaulay type. Adv. Math. 226 (2011), 1973-2019. DOI 10.1016/j.aim.2010.09.006 | MR 2737805 | Zbl 1239.16016
  [4] D. Bennis, N. Mahdou: Strongly Gorenstein projective, injective and flat modules. J. Pure Appl. Algebra 210 (2007), 437-445. DOI 10.1016/j.jpaa.2006.10.010 | MR 2320007 | Zbl 1118.13014
  [5] E. E. Enochs, O. M. G. Jenda: Gorenstein injective and projective modules. Math. Z. 220 (1995), 611-633. DOI 10.1007/BF02572634 | MR 1363858 | Zbl 0845.16005
  [6] E. E. Enochs, O. M. G. Jenda: Relative Homological Algebra. De Gruyter Expositions in Mathematics 30, Walter de Gruyter, Berlin (2000). DOI 10.1515/9783110803662 | MR 1753146 | Zbl 0952.13001
  [7] N. Gao, P. Zhang: Strongly Gorenstein projective modules over upper triangular matrix Artin algebras. Commun. Algebra 37 (2009), 4259-4268. DOI 10.1080/00927870902828934 | MR 2588847 | Zbl 1220.16013
  [8] D. Happel: Triangulated Categories in the Representation Theory of Finite Dimensional Algebras. London Mathematical Society Lecture Note Series 119, Cambridge University Press, Cambridge (1988). DOI 10.1017/CBO9780511629228 | MR 0935124 | Zbl 0635.16017
  [9] H. Holm: Gorenstein homological dimensions. J. Pure Appl. Algebra 189 (2004), 167-193. DOI 10.1016/j.jpaa.2003.11.007 | MR 2038564 | Zbl 1050.16003
  [10] C. Wang: Gorenstein injective modules over upper triangular matrix Artin algebras. J. Shandong Univ., Nat. Sci. 51 (2016), 89-93 (in Chinese). DOI 10.6040/j.issn.1671-9352.0.2015.235 | MR 3467852 | Zbl 06634874
  [11] X. Yang, Z. Liu: Strongly Gorenstein projective, injective and flat modules. J. Algebra 320 (2008), 2659-2674. DOI 10.1016/j.jalgebra.2008.07.006 | MR 2441993 | Zbl 1173.16006
  [12] P. Zhang: Gorenstein-projective modules and symmetric recollements. J. Algebra 388 (2013), 65-80. DOI 10.1016/j.jalgebra.2013.05.008 | MR 3061678 | Zbl 06266167

Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade. To activate your access, please contact Myris Trade at
Subscribers of Springer need to access the articles on their site, which is

Affiliations:   Chao Wang (corresponding author), Xiaoyan Yang, Department of Mathematics, Northwest Normal University, Anning East Road No. 967, Lanzhou, 730070, Gansu, P. R. China, e-mail:,

PDF available at: