On Gorenstein coalgebras

Abstract

We investigate the comodule representation category over the Morita-Takeuchi context coalgebra Γ and study the Gorensteinness of Γ. Moreover, we determine explicitly all Gorenstein injective comodules over the Morita-Takeuchi context coalgebra Γ and discuss the localization in Gorenstein coalgebras. In particular, we describe its Gabriel quiver and carry out some examples when the Morita-Takeuchi context coalgebra is basic.

Keywords

Gorenstein coalgebra / Gorenstein injective comodule / Gabriel quiver

References

[1] Asensio J M, Lopez Ramos A J, Torrecillas B. Gorenstein coalgeras. Acta Math Hungar, 1999, 85: 187–198

[2] Chin W. The translation operator for self-projective coalgebras. Comm Algebra, 2011, 39: 4608–4617

[3] Chin W, Kleiner M, Quinn D. Almost split sequences for comodules. J Algebra, 2002, 249: 1–19

[4] Cuadra J, Gomez-Torrecillas J. Serial coalgebras. J Pure Appl Algebra, 2004, 189: 89–107

[5] Cuadra J, Simson D. Flat comodules and perfect coalgebras. Comm Algebra, 2007, 35: 3164–3194

[6] Dǎscǎlescu S, Nǎstǎsescu C, Raianu S, Van Oystaeyen F. Graded coalgebras and Morita-Takeuchi contexts. Tsukuba J Math, 1995, 19: 395–407

[7] Enochs E, Lopez Ramos A J. Relative homological coalgebras. Acta Math Hungar, 2004, 104(4): 331–343

[8] Fu X, Yao H. The indecomposable injective comodules over Morita-Takeuchi context coalgebras. J Beijing Univ Technol, 2015, 41(9): 1430–1436

[9] Gomez Torrecillas J, Nastasescu C. Quasi-co-Frobenius coalgebras. J Algebra, 1995, 174: 900–923

[10] Green E, Pasroudakis C. On Artin algebras arising from Morita contexts. Algebr Represent Theory, 2014, 17: 1485–1525

[11] Iovanov M C. Triangular matrix coalgebras and applications. Linear Multilinear Algebra, 2015, 63(1): 46–67

[12] Kosakowska J, Simson D. Hereditary coalgebras and representations of species. J Algebra, 2005, 293: 457–505

[13] Kosakowska J, Simson D. Bipartite coalgebras and a reduction functor for coradical square complete coalgebras. Colloq Math, 2008, 112(1): 89–129

[14] Lin I-P. Semiperfect coalgebras. J Algebra, 1977, 49: 357–373

[15] Nastasescu C, Torrecillas B, Zhang Y H. Hereditary coalgebras. Comm Algebra, 1996, 24(4): 1521–1528

[16] Navarro G. Some remarks on localization in coalgebras. Comm Algebra, 2008, 36: 3447–3466

[17] Pan Q, Li Q. On Gorenstein injective and projective comodules. Math Notes, 2013, 94(1): 255–265

[18] Takeuchi M. Morita theorem for categories of comodules. J Fac Sci Univ Tokyo, 1977, 24: 629–644

[19] Tang G, Li C, Zhang Y. Study of Morita contexts. Comm Algebra, 2014, 42: 1668–1681

[20] Wang D. Morita-Takeuchi contexts acting on graded coalgebras. Algebra Colloq, 2000, 7(1): 73–82

[21] Xiong B, Zhang P. Gorenstein-projective modules over triangular matrix Artin algebras. J Algebra Appl, 2012, 11(4): 1–14

[22] Zhang P. Gorenstein-projective modules and symmetric recollements. J Algebra, 2013, 388: 65–80

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg