Geodesic
Elementary equivalence of Chevalley groups over local rings
Sbornik. Mathematics, Tome 201 (2010) no. 3, pp. 321-337 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

It is proved that (elementary) Chevalley groups over local rings with invertible 2 are elementarily equivalent if and only if their types and weight lattices coincide and the initial rings are elementarily equivalent. Bibliography: 25 titles.
Keywords: Chevalley groups, elementary equivalence, local rings. 
@article{SM_2010_201_3_a0,
     author = {E. I. Bunina},
     title = {Elementary equivalence of {Chevalley} groups over local rings},
     journal = {Sbornik. Mathematics},
     pages = {321--337},
     year = {2010},
     volume = {201},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2010_201_3_a0/}
}
TY  - JOUR
AU  - E. I. Bunina
TI  - Elementary equivalence of Chevalley groups over local rings
JO  - Sbornik. Mathematics
PY  - 2010
SP  - 321
EP  - 337
VL  - 201
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/SM_2010_201_3_a0/
LA  - en
ID  - SM_2010_201_3_a0
ER  - 
%0 Journal Article
%A E. I. Bunina
%T Elementary equivalence of Chevalley groups over local rings
%J Sbornik. Mathematics
%D 2010
%P 321-337
%V 201
%N 3
%U http://geodesic.mathdoc.fr/item/SM_2010_201_3_a0/
%G en
%F SM_2010_201_3_a0
E. I. Bunina. Elementary equivalence of Chevalley groups over local rings. Sbornik. Mathematics, Tome 201 (2010) no. 3, pp. 321-337. http://geodesic.mathdoc.fr/item/SM_2010_201_3_a0/

[1] C. C. Chang, H. J. Keisler, Model theory, North-Holland, Amsterdam–London; Elsevier, New York, 1973 | MR | MR | Zbl

[2] A. I. Maltsev, “Ob elementarnykh svoistvakh lineinykh grupp”, Nekotorye problemy matematiki i mekhaniki, Novosibirsk, Izd-vo SO AN SSSR, 1961, 110–132 | MR | Zbl

[3] C. I. Beidar, A. V. Mikhalëv, “On Mal'cev's theorem on elementary equivalence of linear groups”, Proceedings of the International Conference on Algebra (Novosibirsk, 1989), Contemp. Math., 131, Amer. Math. Soc., Providence, RI, 1992, 29–35 | MR | Zbl

[4] E. I. Bunina, “Elementary equivalence of unitary linear groups over rings and skew fields”, Russian Math. Surveys, 53:2 (1998), 374–376 | DOI | MR | Zbl

[5] E. I. Bunina, “Elementary equivalence of Chevalley groups”, Russian Math. Surveys, 56:1 (2001), 152–153 | DOI | MR | Zbl

[6] E. I. Bunina, A. V. Mikhalev, “Combinatorial and logical aspects of linear groups and Chevalley groups”, Acta Appl. Math., 85:1–3 (2005), 57–74 | DOI | MR | Zbl

[7] E. I. Bunina, “Chevalley groups over fields and their elementary properties”, Russian Math. Surveys, 59:5 (2004), 952–953 | DOI | MR | Zbl

[8] E. I. Bunina, “Elementary properties of Chevalley groups over local rings”, Russian Math. Surveys, 61:2 (2006), 349–350 | DOI | MR | Zbl

[9] E. I. Bunina, “Elementary equivalence of Chevalley groups over fields”, J. Math. Sci. (N. Y.), 152:2 (2008), 155–190 | DOI | MR | Zbl

[10] E. I. Bunina, “Automorphisms of Chevalley groups of some types over local rings”, Russian Math. Surveys, 62:5 (2007), 983–984 | DOI | MR | Zbl

[11] E. I. Bunina, “Automorphisms of Chevalley groups of types $B_2$ and $G_2$ over local rings”, J. Math. Sci. (N. Y.), 155:6 (2008), 795–814 | DOI | MR | Zbl

[12] J. E. Humphreys, Introduction to Lie algebras and representation theory, Grad. Texts in Math., 9, Springer-Verlag, New York–Berlin, 1978 | MR | Zbl

[13] N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, 1337, Hermann, Paris, 1968 | MR | MR | Zbl | Zbl

[14] R. Steinberg, Lectures on Chevalley groups, Yale University, New Haven, CT, 1968 | MR | MR | Zbl

[15] C. Chevalley, “Certains schémas de groupes semi-simples”, Séminaire Bourbaki: Vol. 1960/1961, Exposés 205–222, Benjamin, New York–Amsterdam, 1966, 1–16 | MR | Zbl

[16] N. Vavilov, E. Plotkin, “Chevalley groups over commutative rings: I. Elementary calculations”, Acta Appl. Math., 45:1 (1996), 73–113 | DOI | MR | Zbl

[17] H. Matsumoto, “Sur les sous-groupes arithmétiques des groupes semi-simples déployés”, Ann. Sci. École Norm. Sup. (4), 2 (1969), 1–62 | MR | Zbl

[18] E. Abe, “Chevalley groups over local rings”, Tohoku Math. J. (2), 21:3 (1969), 474–494 | DOI | MR | Zbl

[19] M. R. Stein, “Surjective stability in dimension $0$ for $K^2$ and related functors”, Trans. Amer. Soc., 178 (1973), 165–191 | DOI | MR | Zbl

[20] E. Abe, K. Suzuki, “On normal subgroups of Chevalley groups over commutative rings”, Tohoku Math. J. (2), 28:2 (1976), 185–198 | DOI | MR | Zbl

[21] P. M. Cohn, “On the structure of the $GL_2$ of a ring”, Inst. Hautes Études Sci. Publ. Math., 30 (1966), 365–413 | MR | Zbl

[22] R. G. Swan, “Generators and relations for certain special linear groups”, Advances in Math., 6:1 (1971), 1–77 | DOI | MR | Zbl

[23] A. A. Suslin, “One theorem of Cohn”, J. Sov. Math., 17:2 (1981), 1801–1803 | DOI | MR | Zbl | Zbl

[24] E. Abe, “Normal subgroups of Chevalley groups over commutative rings”, Algebraic $K$-theory and algebraic number theory (Honolulu, HI, 1987), Contemp. Math., 83, Amer. Math. Soc., Providence, RI, 1989, 1–17 | MR | Zbl

[25] E. Abe, “Automorphisms of Chevalley groups over commutative rings”, St. Petersburg Math. J., 5:2 (1993), 287–300 | MR | Zbl