Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-25T16:03:32.338Z Has data issue: false hasContentIssue false

2 - τ-tilting Theory – an Introduction

Published online by Cambridge University Press:  25 November 2023

Edited by
David Jordan ,
Nadia Mazza  and
Sibylle Schroll
David Jordan
Affiliation:
University of Edinburgh
Nadia Mazza
Affiliation:
Lancaster University
Sibylle Schroll
Affiliation:
Universität zu Köln
Get access

Summary

The notion of τ-tilting theory was introduced by Adachi, Iyama and Reiten at the beginning of the last decade and quickly became one of the most active areas of research in the representation theory of finite-dimensional algebras. The aim of these notes is two-fold. On the one hand, we want to give a friendly introduction to τ-tilting theory to anyone with a small background in representation theory. On the other, we want to fill the apparent gap for a survey on the subject by collecting in one place many of the most important results in τ-tilting theory.

Keywords

representations of finite-dimensional algebrasτ-tilting theorytorsion pairsstability conditionswall-and-chamber structures
Type
Chapter
Information
Modern Trends in Algebra and Representation Theory , pp. 46 - 108
Publisher: Cambridge University Press
Print publication year: 2023

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

[1]Adachi, T. 2016. Characterizing τ-tilting finite algebras with radical square zero. Proc. Am. Math. Soc., 144(11), 4673–4685.Google Scholar
[2]Adachi, T. The classification of τ-tilting modules over Nakayama algebras. J. Algebra, 452:227–262, 2016.CrossRefGoogle Scholar
[3]Adachi, T., Aihara, T. and Chan, A. Classification of two-term tilting complexes over Brauer graph algebras. Math. Z., 290(1-2):1–36, 2018.CrossRefGoogle Scholar
[4]Adachi, T., Iyama, O. and Reiten, I. τ-tilting theory. Compos. Math., 150(3):415– 452, 2014.CrossRefGoogle Scholar
[5]Aihara, T. and Iyama, O. Silting mutation in triangulated categories. J. Lond. Math. Soc. (2) 85 (2012), 633–668.Google Scholar
[6]Amiot, C. Cluster categories for algebras of global dimension 2 and quivers with potential. Ann. Inst. Fourier, 59(6):2525–2590, 2009.Google Scholar
[7]Angeleri Hu¨gel, L., Happel, D., and Krause, H., ed. 2007. Handbook of Tilting Theory, Volume 332. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
[8]Asai, S. Semibricks. Int. Math. Res. Not., 2020(16):4993–5054, 2020.CrossRefGoogle Scholar
[9]Asai, S. The wall-chamber structures of the real Grothendieck groups. Adv. Math., 381:45, 2021. Id/No 107615.CrossRefGoogle Scholar
[10]Asai, S. and Pfeifer, C. Wide subcategories and lattices of torsion classes. https://arxiv.org/abs/1905.01148.Google Scholar
[11]Assem, I. Torsion theories induced by tilting modules. Canad. J. Math., 36(5):899–913, 1984.CrossRefGoogle Scholar
[12]Assem, I. Tilting theory an introduction. Topics in algebra. Pt. 1: Rings and representations of algebras, Pap. 31st Semester Class. Algebraic Struct., Warsaw/Poland 1988, Banach Cent. Publ. 26, Part 1, 127–180 (1990), 1990.Google Scholar
[13]Assem, I., Bru¨stle, T. and Schiffler, R. Cluster-tilted algebras and slices. J. Algebra, 319(8):3464–3479, 2008.CrossRefGoogle Scholar
[14]Assem, I., Bru¨stle, T. and Schiffler, R. Cluster-tilted algebras as trivial extensions. Bull. Lond. Math. Soc., 40(1):151–162, 2008.CrossRefGoogle Scholar
[15]Assem, I., Bru¨stle, T. and Schiffler, R. On the Galois coverings of a cluster-tilted algebra. J. Pure Appl. Algebra, 213(7):1450–1463, 2009.CrossRefGoogle Scholar
[16]Assem, I., Bru¨stle, T. and Schiffler, R. Cluster-tilted algebras without clusters. J. Algebra, 324(9):2475–2502, 2010.CrossRefGoogle Scholar
[17]Assem, I., Simson, D., and Skowron´ski, A. 2006. Elements of the Representation Theory of Associative Algebras. Vol. 1. London Mathematical Society Student Texts, vol. 65. Cambridge University Press, Cambridge. Techniques of representation theory.CrossRefGoogle Scholar
[18]Assem, I. and Skowron´ski, A. Iterated tilted algebras of type A˜ n. Math. Z. 195, 269–290 (1987; Zbl 0601.16022).CrossRefGoogle Scholar
[19]August, J. 2020. The tilting theory of contraction algebras. Adv. Math., 374, Article 107372.Google Scholar
[20]Auslander, M. Representation theory of Artin algebras. I, II. Comm. Algebra, 1:177–310, 1974.CrossRefGoogle Scholar
[21]Auslander, M., Platzeck, M. I. and Reiten, I. Coxeter functors without diagrams. Trans. Amer. Math. Soc., 250:1–46, 1979.Google Scholar
[22]Auslander, M. and Reiten, I. Representation theory of artin algebras III almost split sequences. Communications in Algebra, 3(3):239–294, 1975.CrossRefGoogle Scholar
[23]Auslander, M. and Reiten, I. Representation theory of Artin algebras. IV: Invariants given by almost split sequences. Commun. Algebra, 5:443–518, 1977.Google Scholar
[24]Auslander, M. and Reiten, I. Modules determined by their composition factors. Illinois J. Math., 29(2):280–301, 1985.CrossRefGoogle Scholar
[25]Auslander, M., Reiten, I., and Smalø, S. O. 1995. Representation Theory of Artin Algebras. Cambridge Studies in Advanced Mathematics, vol. 36. Cambridge University Press, Cambridge.Google Scholar
[26]Auslander, M. and Smalø, S. O. Addendum to “Almost split sequences in subcategories”, J. Algebra, 71:592–594, 1981.CrossRefGoogle Scholar
[27]Auslander, M. and Smalø, S. O. Almost split sequences in subcategories. J. Algebra, 69:426–454, 1981.CrossRefGoogle Scholar
[28]Barnard, E., Carroll, A. and Zhu, S. Minimal inclusions of torsion classes. Algebr. Comb., 2(5):879–901, 2019.CrossRefGoogle Scholar
[29]Baumann, P., Kamnitzer, J. and Tingley, P. Affine Mirkovic´-Vilonen polytopes. Publ. Math. Inst. Hautes E´tudes Sci., 120:113–205, 2014.Google Scholar
[30]Bautista, R. On algebras of strongly unbounded representation type. Comment. Math. Helv., 60(3):392–399, 1985.Google Scholar
[31]Berenstein, A., Fomin, S. and Zelevinsky, A. 2005. Cluster algebras. III. Upper bounds and double Bruhat cells. Duke Math. J., 126(1), 1–52.Google Scholar
[32]Bernsˇte˘ın, I. N., Gelfand, I. M. and Ponomarev, V. A. Coxeter functors, and Gabriel’s theorem. Uspehi Mat. Nauk, 28(2(170)):19–33, 1973.Google Scholar
[33]Bongartz, K. Tilted algebras. Representations of algebras, Proc. 3rd Int. Conf., Puebla/Mex. 1980, Lect. Notes Math. 903, 26–38, 1981.Google Scholar
[34]Brenner, S. and Butler, M. C. R. Generalizations of the Bernstein-Gel′fand-Ponomarev reflection functors. In Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), volume 832 of Lecture Notes in Math., pages 103–169. Springer, Berlin-New York, 1980.CrossRefGoogle Scholar
[35]Bridgeland, T. 2017. Scattering diagrams, Hall algebras and stability conditions. Algebr. Geom., 4(5), 523–561.Google Scholar
[36]Bru¨stle, T.ms, Smith, D. and Treffinger, H. Wall and chamber structure for finite-dimensional algebras. Adv. Math., 354:106746, 31, 2019.CrossRefGoogle Scholar
[37]Bru¨stle, T.ms, Smith, D. and Treffinger, H. Stability Conditions and Maximal Green Sequences in Abelian Categories. https://arxiv.org/abs/1805.04382.Google Scholar
[38]Bru¨stle, T. and Yang, D. Ordered exchange graphs. In Advances in representation theory of algebras. Selected papers of the 15th international conference on representations of algebras and workshop (ICRA XV), Bielefeld, Germany, August 8–17, 2012, pages 135–193. Zu¨rich: European Mathematical Society (EMS), 2014.CrossRefGoogle Scholar
[39]Buan, A. B., Marsh, B. R., Reineke, M., Reiten, I. and Todorov, G. 2006. Tilting theory and cluster combinatorics. Adv. Math., 204(2), 572–618.Google Scholar
[40]Buan, A. B., Marsh, B. R. and Reiten, I. Cluster-tilted algebras. Trans. Amer. Math. Soc., 359(1):323–332 (electronic), 2007.Google Scholar
[41]Buan, A. B., Marsh, B. R., Reiten, I. and Todorov, G. 2007. Clusters and seeds in acyclic cluster algebras. Proc. Amer. Math. Soc., 135(10), 3049–3060.Google Scholar
[42]Buan, A. B. and Zhou, Y. A silting theorem. J. Pure Appl. Algebra, 220(7):2748– 2770, 2016.CrossRefGoogle Scholar
[43]Caldero, P. and Keller, B. 2006. From triangulated categories to cluster algebras. II. Ann. Sci. E´cole Norm. Sup. (4), 39(6), 983–1009.Google Scholar
[44]Dehy, R. and Keller, B. On the combinatorics of rigid objects in 2-Calabi-Yau categories. Int. Math. Res. Not. IMRN, (11):Art. ID rnn029, 17, 2008.Google Scholar
[45]Demonet, L., Iyama, O. and Jasso, G. τ-tilting finite algebras, bricks, and g-vectors. Int. Math. Res. Not. IMRN, (3):852–892, 2019.Google Scholar
[46]Demonet, L., Iyama, O., Reading, N., Reiten, I. and Thomas, H. Lattice theory of torsion classes. arXiv (1711.01785), 2017.Google Scholar
[47]Derksen, H. and Fei, J. General presentations of algebras. Adv. Math., 278:210– 237, 2015.CrossRefGoogle Scholar
[48]Derksen, H., Weyman, J. and Zelevinsky, A. 2008. Quivers with potentials and their representations. I. Mutations. Selecta Math. (N.S.), 14(1), 59–119.Google Scholar
[49]Derksen, H., Weyman, J. and Zelevinsky, A. Quivers with potentials and their representations II: applications to cluster algebras. J. Amer. Math. Soc., 23(3):749–790, 2010.Google Scholar
[50]Dickson, S. E. A torsion theory for Abelian categories. Trans. Amer. Math. Soc., 121:223–235, 1966.CrossRefGoogle Scholar
[51]Fomin, S. and Zelevinsky, A. 2002. Cluster algebras. I. Foundations. J. Amer. Math. Soc., 15(2), 497–529.Google Scholar
[52]Fomin, S. and Zelevinsky, A. 2003. Cluster algebras. II. Finite type classification. Invent. Math., 154(1), 63–121.Google Scholar
[53]Fomin, S. and Zelevinsky, A. 2007. Cluster algebras. IV. Coefficients. Compos. Math., 143(1), 112–164.Google Scholar
[54]Fu, C. c-vectors via τ-tilting theory. J. Algebra, 473:194–220, 2017.CrossRefGoogle Scholar
[55]Gabriel, P. 1972. Unzerlegbare Darstellungen. I. Manuscripta Math., 6, 71–103.Google Scholar
[56]Gabriel, P. Indecomposable representations. II. Sympos. math. 11, Algebra commut., Geometria, Convegni 1971/1972, 81–104 (1973).Google Scholar
[57]Garver, A., McConville, T. and Serhiyenko, K. Minimal length maximal green sequences. Se´m. Lothar. Combin., 78B:Art. 16, 12, 2017.Google Scholar
[58]Garver, A., McConville, T. and Serhiyenko, K. Minimal length maximal green sequences. Adv. in Appl. Math., 96:76–138, 2018.Google Scholar
[59]Gorsky, M., Nakaoka, H. and Palu, Y. Positive and negative extensions in extriangulated categories https://arxiv.org/abs/2103.12482.Google Scholar
[60]Gross, M., Hacking, P., Keel, S. and Kontsevich, M. 2018. Canonical bases for cluster algebras. J. Amer. Math. Soc., 31(2), 497–608.Google Scholar
[61]Gustafson, W. H. The history of algebras and their representations. In Representations of algebras (Puebla, 1980), volume 944 of Lecture Notes in Math., pages 1–28. Springer, Berlin-New York, 1982.Google Scholar
[62]Happel, D. Triangulated categories in the representation theory of finite-dimensional algebras, volume 119 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1988.CrossRefGoogle Scholar
[63]Happel, D. and Ringel, C. M. Tilted algebras. Trans. Amer. Math. Soc., 274(2):399–443, 1982.Google Scholar
[64]Happel, D. and Unger, L. Almost complete tilting modules. Proc. Amer. Math. Soc., 107(3):603–610, 1989.Google Scholar
[65]Igusa, K., Orr, K., Todorov, G. and Weyman, J. Cluster complexes via semi-invariants. Compos. Math., 145(4):1001–1034, 2009.CrossRefGoogle Scholar
[66]Ingalls, C. and Thomas, H. 2009. Noncrossing partitions and representations of quivers. Compos. Math., 145(6), 1533–1562.Google Scholar
[67]Iyama, O. and Reiten, I. Introduction to τ-tilting theory. Proc. Natl. Acad. Sci. USA, 111(27):9704–9711, 2014.Google Scholar
[68]Jasso, G. Reduction of τ-tilting modules and torsion pairs. Int. Math. Res. Not. IMRN, (16):7190–7237, 2015.Google Scholar
[69]Keller, B. On cluster theory and quantum dilogarithm identities. In Representations of Algebras and Related Topics, pages 85–116. European Mathematical Society Publishing House, Zurich, Switzerland, 2011.Google Scholar
[70]Keller, B. and Demonet, L. A survey on maximal green sequences, 2020.Google Scholar
[71]Keller, B. and Vossieck, D. Aisles in derived categories. Bull. Soc. Math. Belg., 40 (1988), 239–253.Google Scholar
[72]Keller, B. and Yang, D. Derived equivalences from mutations of quivers with potential. Adv. Math., 226(3):2118–2168, 2011.Google Scholar
[73]King, A. D. Moduli of representations of finite dimensional algebras. QJ Math, 45(4):515–530, 1994.CrossRefGoogle Scholar
[74]Liu, S. Semi-stable components of an Auslander-Reiten quiver. J. London Math. Soc. (2), 47(3):405–416, 1993.CrossRefGoogle Scholar
[75]Marks, F. and J. Sˇˇtov´ıcˇek. Torsion classes, wide subcategories and localisations. Bull. Lond. Math. Soc., 49(3):405–416, 2017.Google Scholar
[76]Miller, H., editor. Handbook of homotopy theory. Boca Raton, FL: CRC Press, 2020.Google Scholar
[77]Miyashita, Y. Tilting modules of finite projective dimension. Math. Z., 193:113– 146, 1986.Google Scholar
[78]Mizuno, Y. Classifying τ-tilting modules over preprojective algebras of Dynkin type. Math. Z., 277(3-4):665–690, 2014.Google Scholar
[79]Mou, L. Scattering diagrams of quivers with potentials and mutations. https://arxiv.org/abs/1910.13714.Google Scholar
[80]Mousavand, K. τ-tilting finiteness of biserial algebras. https://arxiv.org/abs/1904.11514.Google Scholar
[81]Mousavand, K. τ-tilting finiteness of non-distributive algebras and their module varieties. https://arxiv.org/abs/1910.02251.Google Scholar
[82]Mousavand, K. and Paquette, C. Minimal (τ-)tilting infinite algebras. https://arxiv.org/abs/2103.12700.Google Scholar
[83]Mumford, D. Geometric invariant theory. Springer-Verlag, Berlin-New York, ergebnisse edition, 1965.Google Scholar
[84]Nagao, K. Donaldson-Thomas theory and cluster algebras. Duke Math. J. 162, No. 7, 1313–1367 (2013).Google Scholar
[85]Nakanishi, T. and Zelevinsky, A. On tropical dualities in cluster algebras. In Algebraic groups and quantum groups. International conference on representation theory of algebraic groups and quantum groups ’10, Graduate School of Mathematics, Nagoya University, Nagoya, Japan, August 2–6, 2010, pages 217–226. Providence, RI: American Mathematical Society (AMS), 2012.CrossRefGoogle Scholar
[86]Nakaoka, H. and Palu, Y. Extriangulated categories, Hovey twin cotorsion pairs and model structures. Cah. Topol. Ge´om. Diffe´r. Cate´g., 60(2):117–193, 2019.Google Scholar
[87]Nazarova, L. A. and Ro˘ıter, A. V. Matrix questions and the Brauer-Thrall conjectures on algebras with an infinite number of indecomposable representations. In Representation theory of finite groups and related topics (Proc. Sympos. Pure Math., Vol. XXI, Univ. Wisconsin, Madison, Wis., 1970), pages 111–115, 1971.Google Scholar
[88]Padrol, A., Palu, Y., Pilaud, V. and P.-Plamondon, G. Associahedra for finite type cluster algebras and minimal relations between g-vectors. https://arxiv.org/abs/1906.06861.Google Scholar
[89]Palu, Y., Pilaud, V. and P.-Plamondon, G. Non-kissing and non-crossing complexes for locally gentle algebras. J. Comb. Algebra, 3(4):401–438, 2019.Google Scholar
[90]Plamondon, P.ms-G. τ-tilting finite gentle algebras are representation-finite. Pac. J. Math., 302(2):709–716, 2019.Google Scholar
[91]Reiten, I. The use of almost split sequences in the representation theory of Artin algebras. Representations of algebras, 3rd Int. Conf., Puebla/Mex. 1980, Lect. Notes Math. 944, 29–104 (1982), 1982.Google Scholar
[92]Rickard, J. Morita theory for derived categories. J. London Math. Soc. (2), 39(3):436–456, 1989.Google Scholar
[93]Ro˘ıter, A. V. Unboundedness of the dimensions of the indecomposable representations of an algebra which has infinitely many indecomposable representations. Izv. Akad. Nauk SSSR Ser. Mat., 32:1275–1282, 1968.Google Scholar
[94]Rudakov, A. Stability for an abelian category. J. Algebra, 197(1):231–245, 1997.Google Scholar
[95]Schiffler, R. 2014. Quiver representations. CMS Books in Mathematics/Ouvrages de Mathe´matiques de la SMC. Springer, Cham.Google Scholar
[96]Schofield, A. Semi-invariants of quivers. J. London Math. Soc. (2), 43(3):385– 395, 1991.Google Scholar
[97]Schroll, S. and Treffinger, H. 2022. A τ-tilting approach to the first Brauer-Thrall conjecture. Proc. Amer. Math. Soc., 150(11), 4567–4574.Google Scholar
[98]Schroll, S., Treffinger, H. and Valdivieso, Y. 2021. On band modules and τ-tilting finiteness. Math. Z., 299(3–4), 2405–2417.Google Scholar
[99]Sentieri, F. A brick version of a theorem of Auslander. https://arxiv.org/abs/ 2011.09253.Google Scholar
[100]Skowron´ski, A. Generalized standard Auslander-Reiten components without oriented cycles. Osaka J. Math., 30(3):515–527, 1993.Google Scholar
[101]Skowron´ski, A. Regular Auslander-Reiten components containing directing modules. Proc. Amer. Math. Soc., 120(1):19–26, 1994.Google Scholar
[102]Smalø, S. O. The inductive step of the second Brauer-Thrall conjecture. Canadian J. Math., 32(2):342–349, 1980.Google Scholar
[103]Smalø, S. O. Torsion theories and tilting modules. Bull. London Math. Soc., 16(5):518–522, 1984.Google Scholar
[104]Tattar, A. Torsion pairs and quasi-abelian categories. Algebr. Represent. Theor., 2020.Google Scholar
[105]Treffinger, H. τ-tilting theory and τ-slices. Journal of Algebra, 481:362–392, 2017.Google Scholar
[106]Treffinger, H. An algebraic approach to Harder-Narasimhan filtrations. https://arxiv.org/abs/1810.06322.Google Scholar
[107]Treffinger, H. On sign-coherence of c-vectors. J. Pure Appl. Algebra, 223(6): 2382–2400, 2019.Google Scholar
[108]Yurikusa, T. Wide subcategories are semistable. Doc. Math., 23:35–47, 2018.CrossRefGoogle Scholar
[109]Zhang, X. τ-rigid modules over Auslander algebras. Taiwanese J. Math., 21(4):727–738, 2017.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×