Comparaison des homologies du groupe linéaire et de son algèbre de Lie
Loday, Jean-Louis
Annales de l'Institut Fourier, Tome 37 (1987), p. 167-190 / Harvested from Numdam
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Pour un anneau local R l’homologie du groupe discret GL n (R) a un comportement tout à fait analogue à l’homologie de l’algèbre de Lie gl n (A) lorsque A est une algèbre associative sur un corps de caractéristique zéro. L’objet de cet article est de faire une synthèse (sans démonstration) des résultats connus sur ces groupes d’homologie en exhibant leurs liens avec la K-théorie algébrique, l’homologie cyclique et la cohomologie motivique. On y pose un certain nombre de questions et on propose une définition pour l’analogue additif de la cohomologie motivique.

The homology of the discrete groupe GL n (R) for a local ring R behaves like the homology of the Lie algebra gl n (A) for A an associative algebra over a characteristic zero field. The aim of this article is to survey the known results (without giving any proof) about these homology groups and to connect them with algebraic K-theory cyclic homology and motivic cohomology. Some questions are raised and a definition for an “addivitive motivic cohomology theory” is suggested.

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     author = {Loday, Jean-Louis},
     title = {Comparaison des homologies du groupe lin\'eaire et de son alg\`ebre de Lie},
     journal = {Annales de l'Institut Fourier},
     volume = {37},
     year = {1987},
     pages = {167-190},
     doi = {10.5802/aif.1116},
     mrnumber = {89i:17011},
     zbl = {0619.20025},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/AIF_1987__37_4_167_0}
}

Loday, Jean-Louis. Comparaison des homologies du groupe linéaire et de son algèbre de Lie. Annales de l'Institut Fourier, Tome 37 (1987) pp. 167-190. doi : 10.5802/aif.1116. http://gdmltest.u-ga.fr/item/AIF_1987__37_4_167_0/

[1] A. A. Beilinson, Height pairing between algebraic cycles, Contemporary Mathematics, vol. 67 (1987). | MR 902590 | MR 89g:11052 | Zbl 0624.14005

[2] A. A. Beilinson, R. Mac Pherson and V. Schechtmann. Notes on motivic cohomology, Duke J. Math., 54 (1987), 679-710. | MR 899412 | MR 88f:14021 | Zbl 0632.14010

[3] S. Bloch, The dilogarithm and extensions of Lie algebras, Springer Lecture Notes in Math., 854 (1981), 1-23. | MR 618298 | MR 83b:17010 | Zbl 0469.14009

[4] S. Bloch, Algebraic cycles and higher algebraic K-theory, Adv. Math., 61 (1986), 267-304. | MR 852815 | MR 88f:18010 | Zbl 0608.14004

[5] J.-L. Brylinski, Central localization in Hochschild homology, preprint (1987). | MR 984041 | Zbl 0669.55001

[6] J.-L. Cathelineau, Sur l'homologie de SL2 dans la représentation adjointe, Math Scand. (à paraître).

[7] A. Connes, Non commutative differential geometry, Publ. Math. I.H.E.S., 62 (1986), 257-360. | Numdam | MR 823176 | Zbl 0592.46056

[8] J. L. Dupont and C. H. Sah, Scissors congruences, J. Pure App. Alg., 25 (1982), 159-195. | MR 662760 | MR 84b:53062b | Zbl 0496.52004

[9] B. L. Feigin and B. L. Tsygan, The homology of matrix Lie algebras over rings and the Hochschild homology, Uspehi Mat. Nauk, 38 (1983), 217-218. | MR 695483 | MR 85i:17014 | Zbl 0526.17006 | Zbl 0518.17002

[10] Z. Fiedorowicz and J.-L. Loday, Crossed simplicial groups and their associated homology, preprint 1986, 45p., soumis pour publication. | Zbl 0755.18005

[11] T. Goodwillie, On the general linear group and Hochschild homology, Ann. Math., 121 (1965), 383-407. | MR 86i:18013 | Zbl 0566.20021

[12] T. Goodwillie, Relative algebraic K-theory and cyclic homology, Ann. of Maths, 124 (1986), 347-402. | MR 88b:18008 | Zbl 0627.18004

[13] M. Gros, Quelques résultats sur l'homologie cyclique des algèbres en caractéristique positive, Comptes Rendus Acad. Sc. Paris, 304 (1987), 139-142. | MR 88i:18015 | Zbl 0609.18007

[14] D. Guin, Homologie du groupe linéaire et symboles en K-théorie algébrique, Thèse d'État, Université L. Pasteur, Strasbourg, Mai 1987. | Zbl 0609.18005

[15] D. Guin-Waléry and J.-L. Loday, Obstruction à l'excision en K-théorie algébrique, Proc. Evanston Conf. 1980, Springer Lecture Notes in Math., 854 (1981), 179-216. | MR 82h:18009a | Zbl 0461.18007

[16] A. Haefliger, The homology of nilpotent Lie groups made discrete, Astérisque, 113-114 (1984), 206-211. | MR 85h:22018 | Zbl 0546.55021

[17] W. Van Der Kallen, Homology stability of linear groups, Invent. Math., 60 (1980), 269-295. | MR 82c:18011 | Zbl 0415.18012

[18] W. Van Der Kallen, H. Maazen and J. Stienstra, A presentation for some K2(n,R), Bull. Amer. Math. Soc., 81 (1975), 934-936. | MR 51 #12993 | Zbl 0337.13012

[19] C. Kassel et J.-L. Loday, Extensions centrales d'algèbres de Lie, Ann. Inst. Fourier, 32-4 (1982), 119-142. | Numdam | MR 85g:17004 | Zbl 0485.17006

[20] M. Kervaire, Multiplicateurs de Schur et K-théorie, Essays in Topology and related topics (Mémoires dédiés à G. de Rham), 1970, Springer Verlag, pp. 212-225. | MR 43 #321 | Zbl 0211.32903

[21] M. Kolster, K2 of non commutative local rings, J. Algebra, 95 (1985), 173-200. | MR 86k:16021 | Zbl 0588.16019

[22] S. Lichtenbaum, Values of zeta function at non negative integers, Springer Lecture Notes in Math., 1068 (1984), 127-138. | MR 756089 | Zbl 0591.14014

[23] S. Lichtenbaum, The construction of weight-two arithmetic cohomology, Invent. Math., 88 (1987), 183-215. | MR 88d:14011 | Zbl 0615.14004

[24] J.-L. Loday, K-théorie algébrique et représentations de groups, Ann. Sci. Ec. Norm. Sup. Paris, 9 (1976), 309-377. | Numdam | MR 56 #5686 | Zbl 0362.18014

[25] J.-L. Loday, Symboles en K-théorie algébrique supérieure, Comptes Rendus Acad. Sci. Paris, 292 (1981), 863-866. | MR 82f:13005 | Zbl 0493.18006

[26] J.-L. Loday, On the boundary map K3(Λ/I) → K2(Λ,I), in Algebraic K-theory, Evanston 1980, Springer L.N., 854 (1981), 262-268. | Zbl 0467.18003

[27] J.-L. Loday and C. Procesi, Cyclic homology and lambda operations (in preparation). | Zbl 0719.19002

[28] J.-L. Loday and D. Quillen, Cyclic homology and the Lie algebra homology of matrices, Comment. Math. Helv., 59 (1984), 565-591. | MR 86i:17003 | Zbl 0565.17006

[29] Y. Manin, Correspondences, motifs and monoidal transformations, Mat. Sborn., 77, 119 (1968), 475-507. | MR 41 #3482 | Zbl 0199.24803

[30] J. S. Milne, Values of zeta functions of varieties over finite fields, Amer. J. Math., 108 (1986), 297-360. | MR 87g:14019 | Zbl 0611.14020

[31] J. Milnor, Introduction to algebraic K-theory, Princeton University Press (1970).

[32] C. Ogle, A map from cyclic homology to K-theory, Ph. D. thesis Ohio State University, 1984.

[33] C. Ogle and C. Weibel, Relative algebraic K-theory and cyclic homology, in preparation.

[34] T. Pirashvili, « Plus » construction for Lie algebras, Bull. Acad. Sc. Georgian SSR, 118 n° 2 (1985), 253-256. | MR 87e:18012 | Zbl 0576.55013

[35] D. Quillen, Cohomology of groups, Actes Congrès Intern. Math., t. 2, (1970), 47-51. | MR 58 #7627a | Zbl 0225.18011

[36] U. Rehmann, Zentrale Erweiterungen der spezielle lineare Gruppe eines Schiefkörpers, J. Reine Angewandte Math., 301 (1978), 77-104. | MR 80h:18009 | Zbl 0377.20036

[37] S. Rosset and J. Tate, A reciprocity law for K2-traces, Comment. Math. Helv., 58 (1983), 38-47. | MR 85b:11105 | Zbl 0514.18010

[38] C. H. Sah, Homology of classical Lie groups made discrete, I. Stability theorems and Schur multipliers, Comment. Math. Helv., 61 (1986), 308-347. | MR 87m:22029 | Zbl 0607.57025

[39] A. A. Suslin, Stability in algebraic K-theory, Algebraic K-theory, Proc. Oberwolfach 1980, Springer L. N., 966 (1982), 304-333. | MR 85d:18011 | Zbl 0498.18008

[40] A. A. Suslin, Homology of GLn, characteristic classes and Milnor K-theory, Algebraic K-theory Number theory and Analysis, Proc. Bielefeld 1982, Springer L. N., 1046 (1984), 357-384. | Zbl 0528.18007

[41] A. A. Suslin, K3 d'un corps et groupe de Bloch, (en russe) Proc. Steklov Inst. (à paraître).