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Keywords:
(strongly) Gorenstein injective module; upper triangular matrix Artin algebra; triangulated category; recollement
(strongly) Gorenstein injective module; upper triangular matrix Artin algebra; triangulated category; recollement
Summary:
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 $.
References:
[1] Anderson, F. W., Fuller, K. R.: 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] Auslander, M., Reiten, I., Smalø, S. O.: 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] Beligiannis, A.: 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] Bennis, D., Mahdou, N.: 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] Enochs, E. E., Jenda, O. M. G.: Gorenstein injective and projective modules. Math. Z. 220 (1995), 611-633. DOI 10.1007/BF02572634 | MR 1363858 | Zbl 0845.16005
[6] Enochs, E. E., Jenda, O. M. G.: Relative Homological Algebra. De Gruyter Expositions in Mathematics 30, Walter de Gruyter, Berlin (2000). DOI 10.1515/9783110803662 | MR 1753146 | Zbl 0952.13001
[7] Gao, N., Zhang, P.: 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] Happel, D.: 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] Holm, H.: 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] Wang, C.: Gorenstein injective modules over upper triangular matrix Artin algebras. J. Shandong Univ., Nat. Sci. 51 (2016), 89-93 Chinese. DOI 10.6040/j.issn.1671-9352.0.2015.235 | MR 3467852 | Zbl 06634874
[11] Yang, X., Liu, Z.: 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] Zhang, P.: Gorenstein-projective modules and symmetric recollements. J. Algebra 388 (2013), 65-80. DOI 10.1016/j.jalgebra.2013.05.008 | MR 3061678 | Zbl 06266167
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