Multiple commutators of elementary subgroups: end of the line.(English)Zbl 1451.19004

Let $$A$$, $$B$$, $$C$$, $$D$$ be two-sided ideals in the associative ring with unit $$R$$. Denote the ideal $$AB + BA$$ by $$A \circ B$$. Recall that $$E(n,A)$$ is the subgroup of $$\mathrm{GL}(n,R)$$ generated by elementary matrices $$e_{ij}(x)$$ with $$x \in A$$. The key results in the paper are the following two lemmas, previously known for a class of rings containing all commutative rings.
Lemma 7. Let $$n \geq 3$$. Then $$[[E(n,A),E(n,B)],E(n,C)]=[E(n,A \circ B),E(n,C)].$$
Lemma 8. Let $$n \geq 4$$. Then $$[[E(n,A),E(n,B)],[E(n,C),E(n,D)]]=[E(n,A \circ B),E(n,C \circ D)].$$
There is also
Theorem 2. If $$A+B=R$$ then $$[E(n,A),E(n,B)]=E(n,A \circ B).$$

MSC:

 19B99 Whitehead groups and $$K_1$$ 20H25 Other matrix groups over rings 20G35 Linear algebraic groups over adèles and other rings and schemes
Full Text:

References:

 [1] Bak, A., Nonabelian K-theory: the nilpotent class of $$\operatorname{K}_1$$ and general stability, K-Theory, 4, 363-397 (1991) · Zbl 0741.19001 [2] Bass, H., K-theory and stable algebra, Publ. Math. IHES, 22, 5-60 (1964) · Zbl 0248.18025 [3] Bass, H.; Milnor, J.; Serre, J.-P., Solution of the congruence subgroup problem for $$\operatorname{SL}_n(n \geq 3)$$ and $$\operatorname{Sp}_{2 n}(n \geq 2)$$, Inst. Hautes Études Sci. Publ. Math., 33, 59-133 (1967) [4] Borewicz, Z. I.; Vavilov, N. A., The distribution of subgroups in the full linear group over a commutative ring, Proc. Steklov Inst. Math., 3, 27-46 (1985) · Zbl 0653.20048 [5] Geller, S. C.; Weibel, C. A., Subgroups of elementary and Steinberg groups of congruence level $$I^2$$, J. Pure Appl. Algebra, 35, 123-132 (1985) · Zbl 0552.18006 [6] Gerasimov, V. N., Group of units of a free product of rings, Math. USSR Sb., 134, 1, 42-65 (1989) · Zbl 0634.16003 [7] Hazrat, R.; Stepanov, A.; Vavilov, N.; Zhang, Z., The yoga of commutators, J. Math. Sci., 179, 6, 662-678 (2011) · Zbl 1318.20049 [8] Hazrat, R.; Stepanov, A.; Vavilov, N.; Zhang, Z., Commutator width in Chevalley groups, Note Mat., 33, 1, 139-170 (2013) · Zbl 1294.20059 [9] Hazrat, R.; Stepanov, A.; Vavilov, N.; Zhang, Z., The yoga of commutators, further applications, J. Math. Sci., 200, 6, 742-768 (2014) · Zbl 1316.20055 [10] Hazrat, R.; Vavilov, N.; Zhang, Z., Multiple commutator formulas for unitary groups, Isr. J. Math., 219, 287-330 (2017) · Zbl 1403.20049 [11] Hazrat, R.; Vavilov, N.; Zhang, Z., The commutators of classical groups, J. Math. Sci., 222, 4, 466-515 (2017) · Zbl 1393.20023 [12] Hazrat, R.; Zhang, Z., Generalized commutator formula, Commun. Algebra, 39, 4, 1441-1454 (2011) · Zbl 1231.20049 [13] Hazrat, R.; Zhang, Z., Multiple commutator formula, Isr. J. Math., 195, 481-505 (2013) · Zbl 1292.20054 [14] van der Kallen, W., A group structure on certain orbit sets of unimodular rows, J. Algebra, 82, 363-397 (1983) · Zbl 0518.20035 [15] Lavrenov, A.; Sinchuk, S., A Horrocks-type theorem for even orthogonal $$\operatorname{K}_2$$ (5 Sep. 2019), pp. 1-23 [16] Mason, A. W., On subgroups of $$\operatorname{GL}(n, A)$$ which are generated by commutators. II, J. Reine Angew. Math., 322, 118-135 (1981) · Zbl 0438.20034 [17] Mason, A. W.; Stothers, W. W., On subgroups of $$\operatorname{GL}(n, A)$$ which are generated by commutators, Invent. Math., 23, 327-346 (1974) · Zbl 0278.20045 [18] Stepanov, A.; Vavilov, N., Decomposition of transvections: a theme with variations, K-Theory, 19, 2, 109-153 (2000) · Zbl 0944.20031 [19] Suslin, A. A., The structure of the special linear group over polynomial rings, Math. USSR, Izv., 11, 2, 235-253 (1977) · Zbl 0354.13009 [20] Vaserstein, L. N., On the normal subgroups of the $$\operatorname{GL}_n$$ of a ring, (Algebraic K-Theory. Algebraic K-Theory, Evanston, 1980. Algebraic K-Theory. Algebraic K-Theory, Evanston, 1980, Lecture Notes in Math., vol. 854 (1981), Springer: Springer Berlin et al.), 454-465 [21] Stepanov, A., Structure of Chevalley groups over rings via universal localization, J. Algebra, 450, 522-548 (2016) · Zbl 1337.20057 [22] Vavilov, N., Unrelativised standard commutator formula, J. Math. Sci., 243, 527-534 (2019) · Zbl 1428.20033 [23] Vavilov, N., Commutators of congruence subgroups in the arithmetic case, Zap. Nauč. Semin. POMI, 479, 5-22 (2019) [24] Vavilov, N. A.; Stepanov, A. V., Standard commutator formula, Vestn. St. Petersbg. State Univ., Ser. 1, 41, 1, 5-8 (2008) · Zbl 1147.20042 [25] Vavilov, N. A.; Stepanov, A. V., Standard commutator formulae, revisited, Vestn. St. Petersbg. State Univ., Ser. 1, 43, 1, 12-17 (2010) · Zbl 1254.20038 [26] Vavilov, N.; Zhang, Z., Generation of relative commutator subgroups in Chevalley groups. II, Proc. Edinburgh Math. Soc. (2020) [27] Vavilov, N.; Zhang, Z., Commutators of relative and unrelative elementary groups, revisited (13 Mar. 2020), pp. 1-18 [28] Vavilov, N.; Zhang, Z., Commutators of Relative and Unrelative Elementary Unitary Groups (1 Apr. 2020), pp. 1-40 [29] Weibel, C., $$K_2, K_3$$ and nilpotent ideals, J. Pure Appl. Algebra, 18, 333-345 (1980) · Zbl 0444.13004 [30] Wendt, V., On homotopy invariance for homology of rank two groups, J. Pure Appl. Algebra, 216, 10, 2291-2301 (2012) · Zbl 1267.20062 [31] You, H., On subgroups of Chevalley groups which are generated by commutators, J. Northeast. Normal Univ., 2, 9-13 (1992)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.