Abstract
In this paper we construct Gorenstein-projective modules over Morita rings with zero bimodule homomorphisms and we provide sufficient conditions for such rings to be Gorenstein Artin algebras. This is the first part of our work which is strongly connected with monomorphism categories. In the second part, we investigate monomorphisms where the domain has finite projective dimension. In particular, we show that the latter category is a Gorenstein subcategory of the monomorphism category over a Gorenstein algebra. Finally, we consider the category of coherent functors over the stable category of this Gorenstein subcategory and show that it carries a structure of a Gorenstein abelian category.
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Amiot, C., Iyama, O., Reiten, I.: Stable categories of Cohen-Macaulay modules and cluster categories. Am. J. Math. 137(3), 813–857 (2015)
Auslander, M.: Coherent functors. In: Proceeding Conference Categorical Algebra (La Jolla, Calif., 1965), pp 189–231. Springer, New York (1966)
Auslander, M. Representation dimension of artin algebras, Queen Mary College Notes (1971)
Auslander, M., Reiten, I.: Cohen-Macaulay and Gorenstein Algebras. Progress in Math 95, 221–245 (1991)
Auslander, M., Reiten, I., Smalø, S.: Representation Theory of Artin Algebras. Cambridge University Press (1995)
Bass, H. The Morita theorems, mimeographed notes, University of Oregon (1962)
Beilinson, A., Bernstein, J., Deligne, P.: Faisceaux Pervers, (French) [Perverse sheaves], Analysis and topology on singular spaces, I (Luminy, 1981), 5–171, Asterisque 100 Soc. Math. France, Paris (1982)
Beligiannis, A.: The Homological Theory of Contravariantly Finite Subcategories: Gorenstein Categories, Auslander-Buchweitz Contexts and (Co-)Stabilization. Comm. Algebra 28, 4547–4596 (2000)
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. Amer. Math. Soc. 188(883), viii+207 (2007)
Bennis, D., Mahdou, N.: Strongly Gorenstein projective, injective, and flat modules. J. Pure Appl. Algebra 210, 437–445 (2007)
Buchweitz, R.-O.: Maximal Cohen-Macaulay modules and Tate-Cohomology over Gorenstein rings, unpublished manuscript, p. 155 (1987)
Buchweitz, R.-O.: Morita contexts, idempotents, and Hochschild cohomology - with applications to invariant rings, Contemp. Math. Amer. Math, Soc., Providence, RI 331, 25–53 (2003)
Bühler, T.: Exact categories. Expo. Math. 28(1), 1–69 (2010)
Chen, X.-W.: The stable monomorphism category of a Frobenius category. Math. Res. Lett. 18(1), 125–137 (2011)
Chen, X.-W.: Three results on Frobenius categories. Math. Z. 270(1-2), 43–58 (2012)
Christensen, L.W.: Gorenstein dimensions. In: Lecture Notes in math, vol. 1747. Springer, Berlin (2000)
Cline, E., Parshall, B., Scott, L.: Stratifying endomorphisms algebras. Mem. Amer. Math. Soc. 124(591), viii+119 (1996)
Cohn, P.M. Morita equivalence and duality, Qeen Mary College Math. Notes (1966)
Fossum, R., Griffith, P., Reiten, I.: Trivial Extensions of Abelian Categories with Applications to Ring Theory, vol. 456. Springer L.N.M. (1975)
Franjou, V., Pirashvili, T.: Comparison of abelian categories recollements. Documenta Math. 9, 41–56 (2004)
Green, E.L.: On the representation theory of rings in matrix form. Pacific. J. Math. 100(1), 138–152 (1982)
Green, E.L., Psaroudakis, C.: On Artin algebras arising from Morita contexts. Algebr. Represent. Theory 17(5), 1485–1525 (2014)
Happel, D.: Triangulated categories in the representation theory of finite dimensional algebras, London Math. Soc. Lecture Notes Ser., vol. 119. Cambridge University Press, Cambridge (1988)
Happel, D.: On Gorenstein algebras. In: Representation theory of finite groups and finite-dimensional algebras, Prog. Math., vol. 95, pp 389–404 (1991)
Keller, B.: Chain complexes and stable categories. Manuscripta Math. 67, 379–417 (1990)
Koenig, S., Nagase, H.: Hochschild cohomology and stratifying ideals. J. Pure Appl. Algebra 213(5), 886–891 (2009)
Kussin, D., Lenzing, H., Meltzer, H.: Nilpotent operators and weighted projective lines. J. Reine Angew. Math. 685, 33–71 (2013)
Li, Z.-W., Zhang, P.: A construction of Gorenstein-projective modules. J. Algebra 323(6), 1802–1812 (2010)
Matsui, H., Takahashi, R. Singularity categories and singular equivalences for resolving subcategories, arXiv:1412.8061, Math. Z. (to appear)
McConnell, J.C., Robson, J.C.: Noncommutative Noetherian rings, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication. Wiley, Chichester (1987)
Orlov, D.: Triangulated categories of singularities and D-branes in Landau-Ginzburg models. Tr. Mat. Inst. Steklova 246, 240–262 (2004). English transl.: Proc. Steklov. Inst. Math. 246 (2204), no. 3, 227–248
Psaroudakis, C.: Representation Dimension, Cohen-Macaulay Modules and Triangulated Categories, Ph.D. thesis, p 201. University of Ioannina, Greece (2013)
Psaroudakis, C.: Homological Theory of Recollements of Abelian Categories. J. Algebra 398, 63–110 (2014)
Psaroudakis, C., Vitória, J.: Recollements of Module Categories. Appl. Categ. Structures 22(4), 579–593 (2014)
Psaroudakis, C., Vitória, J. Realisation functors in tilting theory, arXiv:1511.02677
Quillen, D.: Higher algebraic K-theory. I, in Algebraic K-theory, I: higher K-theories, Seattle, WA, 1972. Lecture Notes in Mathematics, vol. 341, pp 85–147. Springer, Berlin (1973)
Ringel, C.M., Schmidmeier, M.: Submodule categories of wild representation type. J. Pure Appl. Algebra 205(2), 412–422 (2006)
Ringel, C.M., Schmidmeier, M.: The Auslander-Reiten translation in submodule categories. Trans. Amer. Math. Soc. 360(2), 691–716 (2008)
Ringel, C.M., Schmidmeier, M.: Invariant subspaces of nilpotent linear operators. I, J. Reine Angew. Math. 614, 1–52 (2008)
Rowen, L.H.: Ring Theory. Student edition. Academic Press, Inc., Doston (1991)
Xi, C.C.: Adjoint functors and representation dimensions. Acta. Math. Sin. 22 (2), 625–640 (2006)
Xiong, B.L., Zhang, P., Zhang, Y.H.: Auslander-Reiten translations in monomorphism categories. Forum Math. 26, 863–912 (2014)
Xiong, B.L., Zhang, P.: Gorenstein-projective modules over triangular matrix Artin algebras. J. Algebra Appl. 11(4), 14 (2012)
Zhang, P.: Gorenstein-projective modules and symmetric recollements. J. Algebra 388, 65–80 (2013)
Zhang, P.: Monomorphism categories, cotilting theory, and Gorenstein-projective modules. J. Algebra 339, 181–202 (2011)
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Presented by Jon F. Carlson.
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Gao, N., Psaroudakis, C. Gorenstein Homological Aspects of Monomorphism Categories via Morita Rings. Algebr Represent Theor 20, 487–529 (2017). https://doi.org/10.1007/s10468-016-9652-1
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DOI: https://doi.org/10.1007/s10468-016-9652-1
Keywords
- Monomorphism categories
- Morita rings
- Homological embeddings
- Gorenstein artin algebras
- Gorenstein-projective modules
- Gorenstein (sub)categories
- Coherent functors