Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal, Vol. 67, No. 4, pp. 1031-1048, 2017

(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 Institute of Mathematics CAS Digital Mathematics Library

References:
[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

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: wangchao0314math@163.com, yangxy@nwnu.edu.cn

Springer subscribers can access the papers on Springer website.

[List of online first articles] [Contents of Czechoslovak Mathematical Journal] [Full text of the older issues of Czechoslovak Mathematical Journal at DML-CZ]

PDF available at:

Springer
for subscribers

···

Institute of Mathematics CAS
free access

···

Digital Mathematics Library
free access