The universality of Hughes-free division rings. (English) Zbl 1512.16023
Let \(R=E*G\) be a crossed product of a division ring \(E\) and a locally indicable group \(G\). It was proved by Ian Hughes in his PhD thesis is that, up to isomorphism, there is at most one Hughes-free division epic division ring containing \(E*G\). This paper makes interesting contributions to the problem of the existence of the Hughes-free division ring of fractions and its relation with the universal ring of fractions. To clarify the expected relation between these objects we reproduce the following conjecture formulated by the author in the paper:
It was already known that part (A) of the conjecture holds for crossed products of a division ring with a locally indicable amenable group, or with a residually-(torsion-free nilpotent) groups or with a free-by-cyclic groups. In this paper it is proved that part \((B)\) of the conjecture also holds for such crossed products.
It is also proved that statements (A) and (B) of the conjecture hold for \(R=E[G]\) the group algebra over the division ring \(E\) and \(G\) a residually-(locally indicable and amenable) group.
A key tool in the paper under review are Sylvester matrix rank functions and their extension to module categories. There are a number of interesting results involving them. For example, a criteria to check the universality of an epic \(R\)-division ring \(\mathcal{D}\) in terms of finitely generated \(R\)-submodule of \(\mathcal{D}\) is given, and this is crucial to prove some of the main results.
Summarizing, this is an excellent paper, quite self-contained, carefully written and plenty of interesting ideas on Sylvester matrix rank functions and their applications to Hughes-free embeddings into division rings.
The interested reader may find it useful to check J. Gräter’s paper [Forum Math. 32, No. 3, 739–772 (2020; Zbl 1484.16030)], for further interesting characterizations of Hughes-free embeddings and a good account on related concepts.
- (A)
- The Hughes-free division \(R\)-ring \(\mathcal{D}_R\) exists and
- (B)
- it is the universal division ring of fractions of \(R\).
It was already known that part (A) of the conjecture holds for crossed products of a division ring with a locally indicable amenable group, or with a residually-(torsion-free nilpotent) groups or with a free-by-cyclic groups. In this paper it is proved that part \((B)\) of the conjecture also holds for such crossed products.
It is also proved that statements (A) and (B) of the conjecture hold for \(R=E[G]\) the group algebra over the division ring \(E\) and \(G\) a residually-(locally indicable and amenable) group.
A key tool in the paper under review are Sylvester matrix rank functions and their extension to module categories. There are a number of interesting results involving them. For example, a criteria to check the universality of an epic \(R\)-division ring \(\mathcal{D}\) in terms of finitely generated \(R\)-submodule of \(\mathcal{D}\) is given, and this is crucial to prove some of the main results.
Summarizing, this is an excellent paper, quite self-contained, carefully written and plenty of interesting ideas on Sylvester matrix rank functions and their applications to Hughes-free embeddings into division rings.
The interested reader may find it useful to check J. Gräter’s paper [Forum Math. 32, No. 3, 739–772 (2020; Zbl 1484.16030)], for further interesting characterizations of Hughes-free embeddings and a good account on related concepts.
Reviewer: Dolors Herbera (Barcelona)
MSC:
| 16S35 | Twisted and skew group rings, crossed products |
| 20F65 | Geometric group theory |
| 12E15 | Skew fields, division rings |
| 16S34 | Group rings |
| 16K40 | Infinite-dimensional and general division rings |
Keywords:
locally indicable groups; crossed products; universal division ring of fractions; Hughes-free division ring; Sylvester matrix rank functionsCitations:
Zbl 1484.16030References:
| [1] | Agol, I., Criteria for virtual fibering, J. Topol., 1, 269-284 (2008) · Zbl 1148.57023 · doi:10.1112/jtopol/jtn003 |
| [2] | Agol, I., The virtual Haken conjecture. With an appendix by Agol, Daniel Groves, and Jason Manning, Doc. Math., 18, 1045-1087 (2013) · Zbl 1286.57019 |
| [3] | Cohn, PM, Theory of General Division Rings. Encyclopedia of Mathematics and its Applications (1995), Cambridge: Cambridge University Press, Cambridge · Zbl 0840.16001 |
| [4] | Cohn, PM, Free Ideal Rings and Localization in General Rings. New Mathematical Monographs 3 (2006), Cambridge: Cambridge University Press, Cambridge · Zbl 1114.16001 · doi:10.1017/CBO9780511542794 |
| [5] | Conrad, P., Right-ordered groups, Michigan Math. J., 6, 267-275 (1959) · Zbl 0099.01703 · doi:10.1307/mmj/1028998233 |
| [6] | Deroin, B., Navas, A., Rivas, C.: Groups, orders, and dynamics. arXiv:1408.5805 |
| [7] | Dicks, W., Herbera, D., Sánchez, J.: On a theorem of Ian Hughes about division rings of fractions. Commun. Algebra 32, 1127-1149 (2004) · Zbl 1084.16018 |
| [8] | Gräter, J., Free division rings of fractions of crossed products of groups with Conradian left-orders, Forum Math., 32, 3, 739-772 (2020) · Zbl 1484.16030 · doi:10.1515/forum-2019-0264 |
| [9] | Higman, G., The units of group-rings, Proc. Lond. Math. Soc., 2, 46, 231-248 (1940) · JFM 66.0104.04 · doi:10.1112/plms/s2-46.1.231 |
| [10] | Howie, J.; Schneebeli, H., Homological and topological properties of locally indicable groups, Manuscripta Math., 44, 71-93 (1983) · Zbl 0533.20022 · doi:10.1007/BF01166075 |
| [11] | Hughes, I.: Division rings of fractions for group rings. Comm. Pure Appl. Math. 181-188,(1970) · Zbl 0214.05401 |
| [12] | Hughes, I., Division rings of fractions for group rings. II, Commun. Pure Appl. Math., 25, 127-131 (1972) · Zbl 0238.16012 · doi:10.1002/cpa.3160250202 |
| [13] | Jaikin-Zapirain, A., The base change in the Atiyah and Lück approximation conjectures, Geom. Funct. Anal., 29, 464-538 (2019) · Zbl 1453.20007 · doi:10.1007/s00039-019-00487-3 |
| [14] | Jaikin-Zapirain, A.: \(L^2\)-Betti numbers and their analogues in positive characteristic, Groups St Andrews 2017 in Birmingham, pp. 346-406, London Math. Soc. Lecture Note Ser., 455. Cambridge Univ. Press, Cambridge (2019) · Zbl 1453.20007 |
| [15] | Jaikin-Zapirain, A.; López-Álvarez, D., The strong Atiyah and Lück approximation conjectures for one-relator groups, Math. Ann., 376, 1741-1793 (2020) · Zbl 1481.20093 · doi:10.1007/s00208-019-01926-0 |
| [16] | Jaikin-Zapirain, A.; Shusterman, M., The Hanna Neumann conjecture for Demushkin Groups, Adv. Math., 349, 1-28 (2019) · Zbl 1436.20048 · doi:10.1016/j.aim.2019.04.013 |
| [17] | Jiang, B., Li, H.: Sylvester rank functions for amenable normal extensions. J. Funct. Anal. 280, 108913 (2021), 47 pp · Zbl 1482.16037 |
| [18] | Kielak, D., Residually finite rationally-solvable groups and virtual fibring, J. Am. Math. Soc., 33, 451-486 (2020) · Zbl 1480.20102 · doi:10.1090/jams/936 |
| [19] | Li, H., Bivariant and extended Sylvester rank functions, J. Lond. Math. Soc., 103, 222-249 (2021) · Zbl 1481.16001 · doi:10.1112/jlms.12372 |
| [20] | Malcev, AI, On the embedding of group algebras in division algebras, Doklady Akad. Nauk SSSR (N.S.), 60, 1499-1501 (1948) · Zbl 0034.30901 |
| [21] | Malcolmson, P., Determining homomorphisms to division rings, J. Algebra, 64, 2, 399-413 (1980) · Zbl 0442.16015 · doi:10.1016/0021-8693(80)90153-2 |
| [22] | Morris, D., Amenable groups that act on the line, Algebr. Geom. Topol., 6, 2509-2518 (2006) · Zbl 1185.20042 · doi:10.2140/agt.2006.6.2509 |
| [23] | Neumann, BH, On ordered division rings, Trans. Amer. Math. Soc., 66, 202-252 (1949) · Zbl 0035.30401 · doi:10.1090/S0002-9947-1949-0032593-5 |
| [24] | Passman, D., Universal fields of fractions for polycyclic group algebras, Glasgow Math. J., 23, 103-113 (1982) · Zbl 0486.16004 · doi:10.1017/S0017089500004869 |
| [25] | Sánchez, J.: On division rings and tilting modules, Ph. D. Thesis, Universitat Autónoma de Barcelona (2008), www.tdx.cat/bitstream/handle/10803/3107/jss1de1.pdf |
| [26] | Strebel, R., Homological methods applied to the derived series of groups, Comment. Math. Helv., 49, 302-332 (1974) · Zbl 0288.20066 |
| [27] | Virili, S., Algebraic entropy of amenable group actions, Math. Z., 291, 1389-1417 (2019) · Zbl 1450.16019 · doi:10.1007/s00209-018-2192-0 |
| [28] | Wise, DT, A note on the vanishing of the 2nd \(L^2\)-Betti number, Proc. Am. Math. Soc., 148, 3239-3244 (2020) · Zbl 1484.20069 · doi:10.1090/proc/14967 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
