Comptes Rendus
Algebra, Representation theory
The size of a stratifying system can be arbitrarily large
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 15-19.

In this short note we construct two families of examples of large stratifying systems in module categories of algebras. The first examples consist on stratifying systems of infinite size in the module category of an algebra A. In the second family of examples we show that the size of a finite stratifying system in the module category of a finite dimensional algebra A can be arbitrarily large in comparison to the number of isomorphism classes of simple A-modules. We note that both families of examples are built using well-established results in higher homological algebra.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.385

Hipolito Treffinger 1

1 Institut de Mathématiques Jussieu - Paris Rive Gauge, Université Paris Cité. Paris, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{CRMATH_2023__361_G1_15_0,
     author = {Hipolito Treffinger},
     title = {The size of a stratifying system can be arbitrarily large},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {15--19},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.385},
     language = {en},
}
TY  - JOUR
AU  - Hipolito Treffinger
TI  - The size of a stratifying system can be arbitrarily large
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 15
EP  - 19
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.385
LA  - en
ID  - CRMATH_2023__361_G1_15_0
ER  - 
%0 Journal Article
%A Hipolito Treffinger
%T The size of a stratifying system can be arbitrarily large
%J Comptes Rendus. Mathématique
%D 2023
%P 15-19
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.385
%G en
%F CRMATH_2023__361_G1_15_0
Hipolito Treffinger. The size of a stratifying system can be arbitrarily large. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 15-19. doi : 10.5802/crmath.385. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.385/

[1] Alexey I. Bondal Representation of associative algebras and coherent sheaves, Math. USSR, Izv., Volume 34 (1990) no. 1, pp. 23-42 | DOI | MR | Zbl

[2] Aslak B. Buan; Bethany R. Marsh τ-exceptional sequences, J. Algebra, Volume 585 (2021), pp. 36-68 | DOI | MR | Zbl

[3] Paula Cadavid; Eduardo do N. Marcos Stratifying systems over hereditary algebras, J. Algebra Appl., Volume 14 (2015) no. 6, 1550093, 10 pages | MR | Zbl

[4] Paula Cadavid; Eduardo do N. Marcos Stratifying systems over the hereditary path algebra with quiver 𝔸 p,q , São Paulo J. Math. Sci., Volume 10 (2016) no. 1, pp. 73-90 | DOI | MR | Zbl

[5] William Crawley-Boevey Exceptional sequences of representations of quivers, Representations of algebras. Proceedings of the sixth international conference on representations of algebras, Carleton University, Ottawa, Ontario, Canada, August 19-22, 1992 (CMS Conference Proceedings), Volume 14, American Mathematical Society, 1993, pp. 117-124 | Zbl

[6] Karin Erdmann Stratifying systems, filtration multiplicities and symmetric groups, J. Algebra Appl., Volume 4 (2005) no. 5, pp. 551-555 | DOI | MR | Zbl

[7] Karin Erdmann; Corina Sáenz On standardly stratified algebras, Commun. Algebra, Volume 31 (2003) no. 7, pp. 3429-3446 | DOI | MR | Zbl

[8] Alexey L. Gorodentsev Exceptional fibre bundles on surfaces with moving anticanonical class, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 52 (1988) no. 4, pp. 740-757 | Zbl

[9] Alexey L. Gorodentsev; Alexei N. Rudakov Exceptional vector bundles on projective spaces, Duke Math. J., Volume 54 (1987), pp. 115-130 | MR | Zbl

[10] Martin Herschend; Osamu Iyama; Steffen Oppermann n-representation infinite algebras, Adv. Math., Volume 252 (2014), pp. 292-342 | DOI | MR | Zbl

[11] Martin Herschend; Peter Jørgensen Classification of higher wide subcategories for higher Auslander algebras of type A, J. Pure Appl. Algebra, Volume 225 (2021) no. 5, 106583, 23 pages | MR | Zbl

[12] Lutz Hille; David Ploog Exceptional sequences and spherical modules for the Auslander algebra of k[x]/(x t ), Pac. J. Math., Volume 302 (2019) no. 2, pp. 599-625 | DOI | MR | Zbl

[13] Osamu Iyama Cluster tilting for higher Auslander algebras, Adv. Math., Volume 226 (2011) no. 1, pp. 1-61 | DOI | MR | Zbl

[14] Gustavo Jasso; Julian Külshammer; Chrysostomos Psaroudakis; Sondre Kvamme Higher Nakayama algebras. I: Construction, Adv. Math., Volume 351 (2019), pp. 1139-1200 | DOI | MR | Zbl

[15] Hagen Meltzer Exceptional sequences for canonical algebras, Arch. Math., Volume 64 (1995) no. 4, pp. 304-312 | DOI | MR | Zbl

[16] Octavio Mendoza Hernández; Hipolito Treffinger Stratifying systems through τ-tilting theory, Doc. Math., Volume 25 (2020), pp. 701-720 | MR | Zbl

[17] Eduardo do N. Marcos; Octavio Mendoza; Corina Sáenz Stratifying systems via relative simple modules, J. Algebra, Volume 280 (2004) no. 2, pp. 472-487 | MR | Zbl

[18] Eduardo do N. Marcos; Octavio Mendoza; Corina Sáenz Stratifying systems via relative projective modules, Commun. Algebra, Volume 33 (2005) no. 5, pp. 1559-1573 | MR | Zbl

[19] Steffen Oppermann; Hugh Thomas Higher-dimensional cluster combinatorics and representation theory, J. Eur. Math. Soc., Volume 14 (2012) no. 6, pp. 1679-1737 | DOI | MR | Zbl

[20] Elin Persson Westin Tilting modules and exceptional sequences for leaf quotients of type A zig-zag algebras, Beitr. Algebra Geom., Volume 61 (2020) no. 2, pp. 189-207 | DOI | MR | Zbl

[21] Claus M. Ringel The braid group action on the set of exceptional sequences of a hereditary Artin algebra, AAbelian group theory and related topics (Oberwolfach, 1993) (Contemporary Mathematics), Volume 171, American Mathematical Society (1994), pp. 339-352 | DOI | MR | Zbl

[22] Alexei N. Rudakov Exceptional collections, mutations and helices, Helices and vector bundles (London Mathematical Society Lecture Note Series), Volume 148, Cambridge University Press, 1990, pp. 1-6 | MR | Zbl

[23] Laertis Vaso n-cluster tilting subcategories for radical square zero algebras (2021) (https://arxiv.org/abs/2105.05830)

Cited by Sources:

Comments - Policy