Abstract
We study Morita rings \(\Lambda _{(\phi ,\psi )}=\left (\begin {array}{cc}A &_{A}N_{B} \\ _{B}M_{A} & B \end {array}\right )\) in the context of Artin algebras from various perspectives. First we study covariantly finite, contravariantly finite, and functorially finite subcategories of the module category of a Morita ring when the bimodule homomorphisms \(\phi \) and \(\psi \) are zero. Further we give bounds for the global dimension of a Morita ring \(\Lambda _{(0,0)}\), as an Artin algebra, in terms of the global dimensions of A and B in the case when both \(\phi \) and \(\psi \) are zero. We illustrate our bounds with some examples. Finally we investigate when a Morita ring is a Gorenstein Artin algebra and then we determine all the Gorenstein-projective modules over the Morita ring \(\Lambda _{\phi ,\psi }\) in case \(A=N=M=B\) and A an Artin algebra.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Amitsur, S.A.: Rings of quotients and Morita contexts. J. Algebra 17, 273–298 (1971)
Auslander, M.: Representation Dimension of Artin Algebras. Queen Mary College Mathematics Notes, London (1971)
Auslander, M., Reiten, I.: Applications of Contravariantly Finite Subcategories. Adv. Math. 86, 111–152 (1991)
Auslander, M., Reiten, I.: Cohen-Macaulay and Gorenstein algebras. Prog. Math. 95, 221–245 (1991)
Auslander, M., Reiten, I.: Homologically finite subcategories, London. Math. Soc. Lect. Note Ser. 168, 1–42 (1992)
Auslander, M., Reiten, I., Smalø, S.: Representation Theory of Artin algebras. Cambridge University Press, Cambridge (1995)
Auslander, M., Smalø, S.: Almost split sequences in subcategories. J. Algebra 69(2), 426–454 (1981)
Bass, H.: The Morita Theorems, Mimeographed Notes. University of Oregon (1962)
Bass, H.: Algebraic K-theory. W. A. Benjamin, Inc., New York-Amsterdam (1968)
Beligiannis, A.: On the relative homology of cleft extensions of rings and Abelian categories. J. Pure Appl. Algebra 150, 237–299 (2000)
Beligiannis, A.: Cohen-Macaulay modules (co)torsion pairs and virtually Gorenstein algebras. J. Algebra 288(1), 137–211 (2005)
Beligiannis, A.: On algebras of finite Cohen-Macaulay type. Adv. Math. 226(2), 1973–2019 (2011)
Beligiannis, A., Reiten, I.: Homological and homotopical aspects of torsion theories. Mem. Am. Math. Soc. 188(883), viii+207 (2007)
Buchweitz, R.-O.: Morita Contexts, Idempotents, and Hochschild Cohomology—with Applications to Invariant Rings, vol. 331, pp. 25–53. Contemp. Math. Amer., Math Soc., Providence (2003)
Cohen, M.: A Morita context related to finite automorphism groups of rings. Pac. J. Math. 98, 37–54 (1982)
Cohn, P.M.: Morita Equivalence and Duality. Qeen Mary College Math. Notes (1966)
Fossum, R., Griffith, P., Reiten, I.: Trivial Extensions of Abelian Categories, Homological Algebra of Trivial Extensions of Abelian Categories with Applications to Ring Theory, vol. 456. Springer L.N.M., Berlin-New York (1975)
Franjou, V., Pirashvili, T.: Comparison of abelian categories recollements. Doc. Math. 9, 41–56 (2004)
Gabriel, P., de la Peña, J.A.: Quotients of representation-finite algebras. Comm. Algebra 15(1–2), 279–307 (1987)
Geigle, W., Lenzing, H.: Perpendicular categories with applications to representations and sheaves. J. Algebra 144(2), 273–343 (1991)
Green, E.: On the representation theory of rings in matrix form. Pac. J. Math. 100(1), 138–152 (1982)
Happel, D.: On Gorenstein algebras. In: Representation Theory of Finite Groups and Finite-Dimensional Algebras (Bielefeld, 1991), vol. 95, pp. 389–404. Progr. Math. Birkhäuser, Basel (1991)
Kuhn, N.J.: The generic representation theory of finite fields: a survey of basic structure. In: Infinite length modules (Bielefeld 1998), pp. 193–212. Trends Math., Birkhauser (2000)
Lam, T.Y.: Lectures on Modules and Rings. Graduate Texts in Mathematics, vol. 189. Springer, New York (1999)
Li, Z.-W., Zhang, P.: A construction of Gorenstein-projective modules. J. Algebra 323(6), 1802–1812 (2010)
Loustaunau, P., Shapiro, J.: Homological dimensions in a Morita context with applications to subidealizers and fixed rings. Proc. Am. Math. Soc. 110(3), 601–610 (1990)
Loustaunau, P., Shapiro, J.: Localization in a Morita context with application to fixed ring. J. Algebra 143(2), 373–387 (1991)
McConnell, J.C., Robson, J.C.: Noncommutative Noetherian Rings. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. Wiley, Chichester (1987)
Muller, B.J.: The quotient category of a Morita context. J. Algebra 28, 389–407 (1974)
Palmer, I.: The global homological dimension of semi-trivial extensions of rings. Math. Scand. 37, 223–256 (1975)
Psaroudakis, C.: Extensions of Abelian Categories via Recollements, preprint. University of Ioannina (2011)
Psaroudakis, C.: Homological Theory of Recollements of Abelian Categories, accepted in Journal of Algebra (2013)
Rowen, L.H., Ring Theory. Student Edition. Academic, Boston (1991)
Smalø, S.: Functorial finite subcategories over triangular matrix rings. Proc. Am. Math. Soc. 111, 651–656 (1991)
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by Jon F. Carlson.
The second author has been co-financed by the European Union (European Social Fund - ESF) and Greek national funds through the Operational Program "Education and Lifelong Learning" of the National Strategic Reference Framework (NSRF) - Research Funding Program: Heracleitus II. Investing in knowledge society through the European Social Fund.
Rights and permissions
About this article
Cite this article
Green, E.L., Psaroudakis, C. On Artin Algebras Arising from Morita Contexts. Algebr Represent Theor 17, 1485–1525 (2014). https://doi.org/10.1007/s10468-013-9457-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-013-9457-4
Keywords
- Morita rings
- Functorially finite subcategories
- Global dimension
- Gorenstein Artin algebras
- Gorenstein-projective modules