Lattice theory of torsion classes: beyond \(\tau\)-tilting theory.(English)Zbl 1533.16015
It is well known that torsion classes are closely related to the study of derived categories and their t-structures.
Partly inspired by the cluster algebras ofS. Fomin andA. Zelevinsky [J. Am. Math. Soc. 15, No. 2, 497–529 (2002;Zbl 1021.16017)], the \(\tau\)-tilting theory [T. Adachi et al., Compos. Math. 150, No. 3, 415–452 (2014;Zbl 1330.16004);L. Demonet et al., Int. Math. Res. Not. 2019, No. 3, 852–892 (2019;Zbl 1485.16013)], provides insight into the structure of torsion classes, but it is generally forced to restrict its attention to functorially finite torsion classes.
In the paper under review, the authors develop methods to understand the whole lattice of torsion classes. These methods also shed new light on certain lattices built from Weyl groups, such as the weak order and Cambrian lattices. They establish a lattice theoretical framework to study the partially ordered set \(\operatorname{tors} A\) of torsion classes over a finite dimensional algebra \(A\), and show that \(\operatorname{tors} A\) is a complete lattice which enjoysvery strong properties, as bialgebraicity and complete semidistributivity. They introduce the brick labelling of its Hasse quiver and use it to study lattice congruences of \(\operatorname{tors} A\). In particular, they give a representation-theoretical interpretation of the so-called forcing order, and prove that \(\operatorname{tors} A\) is completely congruence uniform. When \(I\) is a two-sided ideal of \(A\), \(\operatorname{tors} A/I\) is a lattice quotient of \(\operatorname{tors} A\) which is called an algebraic quotient, and the corresponding lattice congruence is called an algebraic congruence.
The second part of this paper consists in studying algebraic congruences. They characterize the arrows of the Hasse quiver of \(\operatorname{tors} A\) that are contracted by an algebraic congruence in terms of the brick labelling. In the third part, they study in detail the case of preprojective algebras \(\Pi\), for which \(\operatorname{tors} \Pi\) is the Weyl group endowed with the weak order. In particular, they give a new, more representation theoretical proof of the isomorphism between \(\operatorname{tors} kQ\) and the Cambrian lattice when \(Q\) is a Dynkin quiver. They also prove that, in type A, the algebraic quotients of \(\operatorname{tors} \Pi\) are exactly its Hasse-regular lattice quotients.
The article is very well written and contains many general results.
Partly inspired by the cluster algebras ofS. Fomin andA. Zelevinsky [J. Am. Math. Soc. 15, No. 2, 497–529 (2002;Zbl 1021.16017)], the \(\tau\)-tilting theory [T. Adachi et al., Compos. Math. 150, No. 3, 415–452 (2014;Zbl 1330.16004);L. Demonet et al., Int. Math. Res. Not. 2019, No. 3, 852–892 (2019;Zbl 1485.16013)], provides insight into the structure of torsion classes, but it is generally forced to restrict its attention to functorially finite torsion classes.
In the paper under review, the authors develop methods to understand the whole lattice of torsion classes. These methods also shed new light on certain lattices built from Weyl groups, such as the weak order and Cambrian lattices. They establish a lattice theoretical framework to study the partially ordered set \(\operatorname{tors} A\) of torsion classes over a finite dimensional algebra \(A\), and show that \(\operatorname{tors} A\) is a complete lattice which enjoysvery strong properties, as bialgebraicity and complete semidistributivity. They introduce the brick labelling of its Hasse quiver and use it to study lattice congruences of \(\operatorname{tors} A\). In particular, they give a representation-theoretical interpretation of the so-called forcing order, and prove that \(\operatorname{tors} A\) is completely congruence uniform. When \(I\) is a two-sided ideal of \(A\), \(\operatorname{tors} A/I\) is a lattice quotient of \(\operatorname{tors} A\) which is called an algebraic quotient, and the corresponding lattice congruence is called an algebraic congruence.
The second part of this paper consists in studying algebraic congruences. They characterize the arrows of the Hasse quiver of \(\operatorname{tors} A\) that are contracted by an algebraic congruence in terms of the brick labelling. In the third part, they study in detail the case of preprojective algebras \(\Pi\), for which \(\operatorname{tors} \Pi\) is the Weyl group endowed with the weak order. In particular, they give a new, more representation theoretical proof of the isomorphism between \(\operatorname{tors} kQ\) and the Cambrian lattice when \(Q\) is a Dynkin quiver. They also prove that, in type A, the algebraic quotients of \(\operatorname{tors} \Pi\) are exactly its Hasse-regular lattice quotients.
The article is very well written and contains many general results.
Reviewer: Shengyong Pan (Beijing)
MSC:
| 16G10 | Representations of associative Artinian rings |
| 06A07 | Combinatorics of partially ordered sets |
| 20F55 | Reflection and Coxeter groups (group-theoretic aspects) |
Cite
References:
| [1] | Adachi, Takahide, Classification of two-term tilting complexes over Brauer graph algebras, Math. Z., 1-36 (2018) ·Zbl 1433.16010 ·doi:10.1007/s00209-017-2006-9 |
| [2] | Adachi, Takahide, \( \tau \)-tilting theory, Compos. Math., 415-452 (2014) ·Zbl 1330.16004 ·doi:10.1112/S0010437X13007422 |
| [3] | Adaricheva, K. V., Join-semidistributive lattices and convex geometries, Adv. Math., 1-49 (2003) ·Zbl 1059.06003 ·doi:10.1016/S0001-8708(02)00011-7 |
| [4] | Adaricheva, K., Lattice theory: special topics and applications. Vol. 2. Classes of semidistributive lattices, 59-101 (2016), Birkh\"{a}user/Springer, Cham ·Zbl 1477.06043 |
| [5] | Asai, Sota, Semibricks, Int. Math. Res. Not. IMRN, 4993-5054 (2020) ·Zbl 1467.16009 ·doi:10.1093/imrn/rny150 |
| [6] | Auslander, Maurice, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, xiv+425 pp. (1997), Cambridge University Press, Cambridge ·Zbl 0834.16001 |
| [7] | Auslander, M., Almost split sequences in subcategories, J. Algebra, 426-454 (1981) ·Zbl 0457.16017 ·doi:10.1016/0021-8693(81)90214-3 |
| [8] | Barnard, Emily, Minimal inclusions of torsion classes, Algebr. Comb., 879-901 (2019) ·Zbl 1428.05314 ·doi:10.5802/alco.72 |
| [9] | Barnard, Emily, Coxeter-biCatalan combinatorics, J. Algebraic Combin., 241-300 (2018) ·Zbl 1387.05270 ·doi:10.1007/s10801-017-0775-1 |
| [10] | Birkhoff, Garrett, Lattice theory, American Mathematical Society Colloquium Publications, Vol. 25, vi+418 pp. (1979), American Mathematical Society, Providence, R.I. ·Zbl 0505.06001 |
| [11] | Bj\"{o}rner, Anders, Combinatorics of Coxeter groups, Graduate Texts in Mathematics, xiv+363 pp. (2005), Springer, New York ·Zbl 1110.05001 |
| [12] | Buan, A. B., Cluster structures for 2-Calabi-Yau categories and unipotent groups, Compos. Math., 1035-1079 (2009) ·Zbl 1181.18006 ·doi:10.1112/S0010437X09003960 |
| [13] | Chajda, I., Advances in algebra. Sectionally pseudocomplemented lattices and semilattices, 282-290 (2003), World Sci. Publ., River Edge, NJ ·Zbl 1048.06006 |
| [14] | P. Crawley and R. P. Dilworth, Algebraic theory of lattices, Prentice-Hall, Englewood Cliffs, NJ, 1973. ·Zbl 0494.06001 |
| [15] | Demonet, Laurent, \( \tau \)-tilting finite algebras, bricks, and \(g\)-vectors, Int. Math. Res. Not. IMRN, 852-892 (2019) ·Zbl 1485.16013 ·doi:10.1093/imrn/rnx135 |
| [16] | Derksen, Harm, General presentations of algebras, Adv. Math., 210-237 (2015) ·Zbl 1361.16006 ·doi:10.1016/j.aim.2015.03.012 |
| [17] | Eisele, Florian, A reduction theorem for \(\tau \)-rigid modules, Math. Z., 1377-1413 (2018) ·Zbl 1433.16011 ·doi:10.1007/s00209-018-2067-4 |
| [18] | Fomin, Sergey, Cluster algebras. I. Foundations, J. Amer. Math. Soc., 497-529 (2002) ·Zbl 1021.16017 ·doi:10.1090/S0894-0347-01-00385-X |
| [19] | Garver, Alexander, Lattice properties of oriented exchange graphs and torsion classes, Algebr. Represent. Theory, 43-78 (2019) ·Zbl 1408.16011 ·doi:10.1007/s10468-017-9757-1 |
| [20] | Gr\"{a}tzer, George, Lattice theory: foundation, xxx+613 pp. (2011), Birkh\"{a}user/Springer Basel AG, Basel ·Zbl 1233.06001 ·doi:10.1007/978-3-0348-0018-1 |
| [21] | Happel, Dieter, Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, x+208 pp. (1988), Cambridge University Press, Cambridge ·Zbl 0635.16017 ·doi:10.1017/CBO9780511629228 |
| [22] | Hoang, Hung P., Combinatorial generation via permutation languages. II. Lattice congruences, Israel J. Math., 359-417 (2021) ·Zbl 1479.05182 ·doi:10.1007/s11856-021-2186-1 |
| [23] | Ingalls, Colin, Noncrossing partitions and representations of quivers, Compos. Math., 1533-1562 (2009) ·Zbl 1182.16012 ·doi:10.1112/S0010437X09004023 |
| [24] | Iyama, Osamu, Lattice structure of Weyl groups via representation theory of preprojective algebras, Compos. Math., 1269-1305 (2018) ·Zbl 1443.16016 ·doi:10.1112/s0010437x18007078 |
| [25] | Iyama, Osamu, Lattice structure of torsion classes for path algebras, Bull. Lond. Math. Soc., 639-650 (2015) ·Zbl 1397.16011 ·doi:10.1112/blms/bdv041 |
| [26] | Iyama, Osamu, Classifying \(\tau \)-tilting modules over the Auslander algebra of \(K[x]/(x^n)\), J. Math. Soc. Japan, 731-764 (2020) ·Zbl 1505.16011 ·doi:10.2969/jmsj/75117511 |
| [27] | Jasso, Gustavo, Reduction of \(\tau \)-tilting modules and torsion pairs, Int. Math. Res. Not. IMRN, 7190-7237 (2015) ·Zbl 1357.16028 ·doi:10.1093/imrn/rnu163 |
| [28] | Kase, Ryoichi, Weak orders on symmetric groups and posets of support \(\tau \)-tilting modules, Internat. J. Algebra Comput., 501-546 (2017) ·Zbl 1384.16011 ·doi:10.1142/S0218196717500266 |
| [29] | Keimel, Klaus, Lattice theory: special topics and applications. Vol. 1. Continuous and completely distributive lattices, 5-53 (2014), Birkh\"{a}user/Springer, Cham ·Zbl 1346.06005 |
| [30] | Mizuno, Yuya, Classifying \(\tau \)-tilting modules over preprojective algebras of Dynkin type, Math. Z., 665-690 (2014) ·Zbl 1355.16008 ·doi:10.1007/s00209-013-1271-5 |
| [31] | J. B. Nation, Revised notes on lattice theory, Lecture Notes available at http://www.math.hawaii.edu/ jb/books.htmlhttp://www.math.hawaii.edu/\( \sim\) jb/books.html. |
| [32] | Palu, Yann, Non-kissing complexes and tau-tilting for gentle algebras, Mem. Amer. Math. Soc., vii+110 pp. (2021) ·Zbl 1541.16001 ·doi:10.1090/memo/1343 |
| [33] | Reading, Nathan, Lattice congruences, fans and Hopf algebras, J. Combin. Theory Ser. A, 237-273 (2005) ·Zbl 1133.20027 ·doi:10.1016/j.jcta.2004.11.001 |
| [34] | Reading, Nathan, Sortable elements and Cambrian lattices, Algebra Universalis, 411-437 (2007) ·Zbl 1184.20038 ·doi:10.1007/s00012-007-2009-1 |
| [35] | Reading, Nathan, Noncrossing arc diagrams and canonical join representations, SIAM J. Discrete Math., 736-750 (2015) ·Zbl 1314.05015 ·doi:10.1137/140972391 |
| [36] | Reading, N., Lattice theory: special topics and applications. Vol. 2. Lattice theory of the poset of regions, 399-487 (2016), Birkh\"{a}user/Springer, Cham ·Zbl 1404.06004 |
| [37] | Reading, N., Lattice theory: special topics and applications. Vol. 2. Finite Coxeter groups and the weak order, 489-561 (2016), Birkh\"{a}user/Springer, Cham ·Zbl 1388.20056 |
| [38] | Ringel, Claus Michael, Representations of \(K\)-species and bimodules, J. Algebra, 269-302 (1976) ·Zbl 0338.16011 ·doi:10.1016/0021-8693(76)90184-8 |
| [39] | Ringel, Claus Michael, Algebras and modules, II. The preprojective algebra of a quiver, CMS Conf. Proc., 467-480 (1996), Amer. Math. Soc., Providence, RI ·Zbl 0928.16012 |
| [40] | Silver, L., Noncommutative localizations and applications, J. Algebra, 44-76 (1967) ·Zbl 0173.03305 ·doi:10.1016/0021-8693(67)90067-1 |
| [41] | Stenstr\"{o}m, Bo, Rings of quotients, Die Grundlehren der mathematischen Wissenschaften, Band 217, viii+309 pp. (1975), Springer-Verlag, New York-Heidelberg ·Zbl 0296.16001 |
| [42] | Storrer, Hans H., Epimorphic extensions of non-commutative rings, Comment. Math. Helv., 72-86 (1973) ·Zbl 0258.16027 ·doi:10.1007/BF02566112 |
| [43] | Wald, Burkhard, Tame biserial algebras, J. Algebra, 480-500 (1985) ·Zbl 0567.16017 ·doi:10.1016/0021-8693(85)90119-X |
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