Hostname: page-component-745bb68f8f-v2bm5 Total loading time: 0 Render date: 2025-01-10T23:22:36.428Z Has data issue: false hasContentIssue false
  • English
  • Français

KOSZUL DUALITY FOR STRATIFIED ALGEBRAS II. STANDARDLY STRATIFIED ALGEBRAS

Part of: Representation theory of rings and algebras Rings and algebras arising under various constructions

Published online by Cambridge University Press:  21 September 2010

VOLODYMYR MAZORCHUK
VOLODYMYR MAZORCHUK*
Affiliation:
Department of Mathematics, Uppsala University, SE 471 06, Uppsala, Sweden (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give a complete picture of the interaction between the Koszul and Ringel dualities for graded standardly stratified algebras (in the sense of Cline, Parshall and Scott) admitting linear tilting (co)resolutions of standard and proper costandard modules. We single out a certain class of graded standardly stratified algebras, imposing the condition that standard filtrations of projective modules are finite, and develop a tilting theory for such algebras. Under the assumption on existence of linear tilting (co)resolutions we show that algebras from this class are Koszul, that both the Ringel and Koszul duals belong to the same class, and that these two dualities on this class commute.

Keywords

stratified algebraKoszul algebratilting moduleRingel duality

MSC classification

Secondary: 16S37: Quadratic and Koszul algebras 16G10: Representations of Artinian rings
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 89 , Issue 1 , August 2010 , pp. 23 - 49
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[1]Ágoston, I., Dlab, V. and Lukács, E., ‘Quasi-hereditary extension algebras’, Algebr. Represent. Theory 6(1) (2003), 97–117.CrossRefGoogle Scholar
[2]Ágoston, I., Dlab, V. and Lukács, E., ‘Standardly stratified extension algebras’, Comm. Algebra 33(5) (2005), 1357–1368.CrossRefGoogle Scholar
[3]Ágoston, I., Happel, D., Lukács, E. and Unger, L., ‘Standardly stratified algebras and tilting’, J. Algebra 226(1) (2000), 144–160.CrossRefGoogle Scholar
[4]Auslander, M. and Buchweitz, R.-O., ‘The homological theory of maximal Cohen–Macaulay approximations’, in: Colloque en l’honneur de Pierre Samuel (Orsay, 1987), Mém. Soc. Math. Fr. (N.S.), 38 (1989), 5–37.Google Scholar
[5]Auslander, M. and Reiten, I., ‘Applications of contravariantly finite subcategories’, Adv. Math. 86(1) (1991), 111–152.CrossRefGoogle Scholar
[6]Chuang, J. and Turner, W., ‘Cubist algebras’, Adv. Math. 217(4) (2008), 1614–1670.CrossRefGoogle Scholar
[7]Cline, E., Parshall, B. and Scott, L., ‘The homological dual of a highest weight category’, Proc. Lond. Math. Soc. (3) 68(2) (1994), 294–316.CrossRefGoogle Scholar
[8]Cline, E., Parshall, B. and Scott, L., ‘Stratifying endomorphism algebras’, Mem. Amer. Math. Soc. 124(591) (1996).Google Scholar
[9]Dlab, V., ‘Quasi-hereditary algebras revisited’, in Representation Theory of Groups, Algebras, and Orders (Constanta, 1995), An. Ştiinţ. Univ. ‘Ovidius’ Constanţa Ser. Mat. 4(2) (1996), 43–54.Google Scholar
[10]Drozd, Y. and Mazorchuk, V., ‘Koszul duality for extension algebras of standard modules’, J. Pure Appl. Algebra 211(2) (2007), 484–496.CrossRefGoogle Scholar
[11]Frisk, A., ‘Two-step tilting for standardly stratified algebras’, Algebra Discrete Math. 3(3) (2004), 38–59.Google Scholar
[12]Frisk, A., ‘Dlab’s theorem and tilting modules for stratified algebras’, J. Algebra 314(2) (2007), 507–537.CrossRefGoogle Scholar
[13]Frisk, A., ‘Typical blocks of the category 𝒪 for the queer Lie superalgebra’, J. Algebra Appl. 6(5) (2007), 731–778.CrossRefGoogle Scholar
[14]Keller, B., ‘Deriving DG categories’, Ann. Sci. École Norm. Sup. (4) 27(1) (1994), 63–102.CrossRefGoogle Scholar
[15]Khomenko, O., Koenig, S. and Mazorchuk, V., ‘Finitistic dimension and tilting modules for stratified algebras’, J. Algebra 286(2) (2005), 456–475.CrossRefGoogle Scholar
[16]Mazorchuk, V., ‘Koszul duality for stratified algebras I. Balanced quasi-hereditary algebras’, Manuscripta Math. 131 (2010), 1–10.CrossRefGoogle Scholar
[17]Mazorchuk, V. and Ovsienko, S., ‘A pairing in homology and the category of linear complexes of tilting modules for a quasi-hereditary algebra’, J. Math. Kyoto Univ. 45(4) (2005), 711–741; With an appendix by C. Stroppel.Google Scholar
[18]Mazorchuk, V., Ovsienko, S. and Stroppel, C., ‘Quadratic duals, Koszul dual functors, and applications’, Trans. Amer. Math. Soc. 361 (2009), 1129–1172.CrossRefGoogle Scholar
[19]Miemietz, V. and Turner, W., ‘Rational representations of GL2’, Glasg. Math. J., to appear.Google Scholar
[20]Rickard, J., ‘Derived equivalences as derived functors’, J. Lond. Math. Soc. (2) 43(1) (1991), 37–48.CrossRefGoogle Scholar
[21]Ringel, C., ‘The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences’, Math. Z. 208(2) (1991), 209–223.CrossRefGoogle Scholar