Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-19T06:23:17.778Z Has data issue: false hasContentIssue false
  • English
  • Français

A GENERALIZATION OF THE THEORY OF STANDARDLY STRATIFIED ALGEBRAS I: STANDARDLY STRATIFIED RINGOIDS

Part of: Representation theory of rings and algebras Abelian categories

Published online by Cambridge University Press:  07 October 2020

O. MENDOZA ,
M. ORTÍZ ,
C. SÁENZ  and
V. SANTIAGO
O. MENDOZA
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico City, Mexico, Circuito Exterior, Ciudad Universitaria, C.P. 04510, Mexico City, D.F. Mexico, e-mail: omendoza@matem.unam.mx
M. ORTÍZ
Affiliation:
Facultad de Ciencias, Universidad Autónoma del Estado de México, Mexico City, Mexico, e-mail: mortizmo@uaemex.mx
C. SÁENZ
Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Mexico City, Mexico, Circuito Exterior, Ciudad Universitaria, C.P. 04510, Mexico City, D.F. Mexico, e-mails: corina.saenz@gmail.com, valente.santiago.v@gmail.com
V. SANTIAGO
Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Mexico City, Mexico, Circuito Exterior, Ciudad Universitaria, C.P. 04510, Mexico City, D.F. Mexico, e-mails: corina.saenz@gmail.com, valente.santiago.v@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We extend the classical notion of standardly stratified k-algebra (stated for finite dimensional k-algebras) to the more general class of rings, possibly without 1, with enough idempotents. We show that many of the fundamental results, which are known for classical standardly stratified algebras, can be generalized to this context. Furthermore, new classes of rings appear as: ideally standardly stratified and ideally quasi-hereditary. In the classical theory, it is known that quasi-hereditary and ideally quasi-hereditary algebras are equivalent notions, but in our general setting, this is no longer true. To develop the theory, we use the well-known connection between rings with enough idempotents and skeletally small categories (ringoids or rings with several objects).

MSC classification

Primary: 18E10: Exact categories, abelian categories
Secondary: 16G99: None of the above, but in this section
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 1 , January 2022 , pp. 1 - 36
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ágoston, I., Dlab, V. and Lukács, E., Stratified algebras, Math. Rep. Acad. Sci. Canada 20(1) (1998), 22–28.Google Scholar
Ágoston, I., Dlab, V. and Lukács, E.. Standardly stratified extensions algebras, Comm. Algebra 33 (2005), 1357–1368.CrossRefGoogle Scholar
Ágoston, I., Dlab, V. and Lukács, E., Approximations of algebras by standardly stratified algebras, J. Algebra 319 (2008), 4177–4198.CrossRefGoogle Scholar
Ágoston, I., Happel, D., Lukács, E. and Unger, L., Standardly stratified algebras and tilting, J. Algebra 226(1) (2000), 144–160.CrossRefGoogle Scholar
Ágoston, I., Happel, D., Lukács, E. and Unger, L., Finitistic dimension of standardly stratified algebras, Comm. Algebra 28(6) (2000), 2745–2752.CrossRefGoogle Scholar
Auslander, M., A functorial approach to representation theory. Representations of algebras, Lecture Notes in Mathematics, vol. 944 (Springer-Verlag, Berlin, New York, 1982), 105–179.Google Scholar
Auslander, M., Representation theory of artin algebras I, Comm. Algebra 3(1) (1974), 177–268.Google Scholar
Bongartz, K. and Gabriel, P., Covering spaces in representation theory, Invent. Math. 65(3) (1982), 331–378.CrossRefGoogle Scholar
Cline, E., Parshall, B. and Scott, L., Derived categories, quasi-hereditary algebras and algebraic groups, in Proceedings of the Ottawa-Moosonee Workshop in Algebra, 1987, Mathematics Lecture Notes Series (Carleton University and Universite d’Ottawa, 1988).Google Scholar
Cline, E., Parshall, B. and Scott, L., Stratifying endomorphism algebras, Mem. AMS 591 (1996), 1–135.Google Scholar
Cline, E., Parshall, B. and Scott, L., Algebraic stratification in representation categories, J. Algebra 117(2) (1988), 504–521.CrossRefGoogle Scholar
De la Peña, J. A. and Martinez, R., The universal cover of a quiver with relations, J. Pure Appl. Algebra 30 (1983), 277–292.Google Scholar
Dlab, V., Quasi-hereditary algebras, in Finite dimensional algebras (Drozd, Yu. A. and Kirichenko, V., Editors) (Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, Hong Kong, Barcelona, Budapest, 1993).Google Scholar
Dlab, V., Quasi-hereditary algebras revisited, An. St. Univ. Ovidius Constanta 4 (1996), 43–54.Google Scholar
Dlab, V., Properly stratified algebras, C. R. Acad. Sci. Paris Sér. I Math. 331(3) (2000), 191–196.CrossRefGoogle Scholar
Dlab, V. and Ringel, C. M., Quasi-hereditary algebras, Illinois J. Math. 33(2) (1989), 280–291.CrossRefGoogle Scholar
Dlab, V. and Ringel, C. M., The module theoretical approach to quasi-hereditary algebras, in Repr. Theory and Related Topics, London Mathematical Society (LMS), vol. 168 (1992), 200–224.Google Scholar
Erdmann, K. and Sáenz, C., On standardly stratified algebras, Comm. Algebra 32 (2003), 3429–3446.CrossRefGoogle Scholar
Freyd, P., Representations in abelian category, in Proceedings of the Conference on Categorical Algebra. La Jolla (1966), 95–120.CrossRefGoogle Scholar
Frisk, A., Two-step tilting for standardly stratified algebras, Algebras Discrete Math. 3 (2004), 38–59.Google Scholar
Frisk, A., Dlab’s theorem and tilting modules for stratified algebras, J. Algebra 314(2) (2007), 507–537.CrossRefGoogle Scholar
Frisk, A. and Mazorchuk, V., Properly stratified algebras and tilting, Proc. London Math. Soc. A (3) 92(1) (2006), 29–61.CrossRefGoogle Scholar
Futorny, V., Konig, S. and Mazorchuk, V., Categories of induced modules and standarly stratified algebras, Algebra Represent. Theory 5(3) (2002), 259–276.CrossRefGoogle Scholar
Gabriel, P., Des catégories abéliennes, Bulletin de la S. M. F. tome 90 (1962), 323–448.Google Scholar
Heller, A., Homological algebra in abelian categories, Ann. Math. Second Ser. 68(3) (1958), 484–525.CrossRefGoogle Scholar
Krause, H., Krull-Schmidt categories and projective covers, Expo. Math. 33(4) (2015), 535–549.CrossRefGoogle Scholar
Krause, H., Highest weight categories and recollements, Annales de l’Institut Fourier tome 67(6) (2017), 2679–2701.CrossRefGoogle Scholar
Marcos, E. N., Mendoza, O. and Sáenz, C., Stratifying systems via relative simple modules, J. Algebra 280 (2004), 472–487.CrossRefGoogle Scholar
Marcos, E: N., Mendoza, O. and Sáenz, C., Stratifying systems via relative projective modules, Comm. Algebra 33 (2005), 1559–1573.CrossRefGoogle Scholar
Martnez-Villa, R. and Ortz-Morales, M., Tilting theory and functor categories II. Generalized Tilting, Appl. Categorical Struct. 21 (2013), 311–348.CrossRefGoogle Scholar
Martnez-Villa, R. and Ortz-Morales, M., Tilting theory and functor categories I. Classical Tilting, Appl. Categorical Struct. 22 (2014), 595–646.CrossRefGoogle Scholar
Martnez-Villa, R. and Solberg, Ø., Graded and Koszul categories, Appl. Categorical Struct. 18 (2010), 615–652.CrossRefGoogle Scholar
Martnez-Villa, R. and Solberg, Ø., Artin-Schelter regular algebras and categories, J. Pure Appl. Algebra 215 (2011), 546–565.CrossRefGoogle Scholar
Martnez-Villa, R. and Solberg, Ø., Noetherianity and Gelfand-Kirilov dimension of components, J. Algebra 323(5) (2010), 1369–1407.CrossRefGoogle Scholar
Mazorchuck, V., Stratified algebras arising in Lie Theory, in Representation of finite dimensional algebras and related topics in lie theory and geometry, Fields Institute Communications, vol. 40 (American Mathematical Society, Providence, RI, 2004), 245–260.Google Scholar
Mazorchuk, V., On the finitistic dimension of stratified algebras, Algebra Discrete Math. 2004(3) (2004), 77–88.Google Scholar
Mazorchuk, V., Koszul duality for stratified algebras II. Standardly stratified algebras, J. Aust. Math. Soc. 89(1) (2010), 23–49.CrossRefGoogle Scholar
Mazorchuk, V. and Ovsienko, S., Finitistic dimension of properly stratified algebras, Adv. Math. 186(1) (2004), 251–265.CrossRefGoogle Scholar
Mazorchuk, V. and Parker, A., On the relation between finitistc and good filtration dimensions, Comm. Algebra 32(5) (2004), 1903–1917.CrossRefGoogle Scholar
Mendoza, O., Sáenz, C. and Xi, C., Homological systems in module categories over pre-ordered sets, Quart. J. Math. 60 (2009), 75–103.CrossRefGoogle Scholar
Mendoza, O. and Santiago, V., Homological systems in triangulated categories, Appl. Categor. Struct. (2014). doi: 10.1007/s10485-014-9384-5.Google Scholar
Mitchell, B., Rings with several objects, Adv. Math. 8 (1972), 1–161.CrossRefGoogle Scholar
Ortiz, M., The Auslander-Reiten components seen as quasi-hereditary categories, M. Appl. Categor. Struct. (2017). https://doi.org/10.1007/s10485-017-9493-z.Google Scholar
Platzeck, M. I. and Reiten, I., Modules of finite projective dimension for standardly stratified algebras, Comm. Algebra 29 (2001), 973–986.CrossRefGoogle Scholar
Ringel, C. M., Representation of K-species and bimodules, J. Algebra 41 (1976), 269–302.CrossRefGoogle Scholar
Ringel, C. M., The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences, Math. Z. 208 (1991), 209–223.CrossRefGoogle Scholar
Scott, L. L., Simulating algebraic geometry with algebra I: The algebraic theory of derived categories, in The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), Proceedings of Symposia in Pure Mathematics, vol. 47 (AMS, 1987), 71–281.CrossRefGoogle Scholar
Webb, P., Standard Stratifications of EI categories and Alperin’s weight conjecture, J. Algebra 320(12) (2008), 4073–4091.CrossRefGoogle Scholar
Wisbauer, R., Foundations of Module and Ring Theory. A Handbook for Study and Research (University of Dusseldorf, Gordon and Breach Science Publishers, Reading, 1991).Google Scholar
Xi, C., Standardly stratified algebras and cellular algebras, Math. Proc. Cambr. Phil. Soc. 133 37–53, (2002).CrossRefGoogle Scholar