Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Curves on $ K$-theory and the de Rham homology of associative algebras


Author: John G. Ryan
Journal: Trans. Amer. Math. Soc. 321 (1990), 559-582
MSC: Primary 19D55; Secondary 16E20, 18G50, 58A12
DOI: https://doi.org/10.1090/S0002-9947-1990-0978383-1
MathSciNet review: 978383
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper describes the generalization to arbitrary associative algebras of the complex of "typical curves on algebraic $ K$-theory" and shows, in particular, that for certain $ {\mathbf{Q}}$-algebras, $ A$, the complex is isomorphic to the "generalized de Rham complex," $ (H{H_*}(A),B)$, in which $ B$ is Connes' operator acting on the Hochschild homology groups of $ A$.


References [Enhancements On Off] (What's this?)

  • [Bl] S. Bloch, Algebraic $ K$-theory and crystalline cohomology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 187-268. MR 488288 (81j:14011)
  • [Bo] N. Bourbaki, Algèbre, Chap. 10: Algèbre homologique, Masson, Paris, 1980. MR 610795 (82j:18022)
  • [Bur-O] D. Burghelea and C. Ogle, The Künneth formula in cyclic homology, Math. Z. 193 (1986), 527-536. MR 867346 (88a:18015)
  • [GRW] S. Geller, L. Reid, and C. Weibel, The cyclic homology and $ K$-theory of curves, J. Reine Angew. Math. (to appear). MR 972360 (89m:14006)
  • [Go 1] T. Goodwillie, Relative algebraic $ K$-theory and cyclic homology, Ann. of Math. (2) 124 (1986), 347-402. MR 855300 (88b:18008)
  • [Go 2] -, On the general linear group and Hochschild homology, Ann. of Math. (2) 121 (1985), 383-407. MR 786354 (86i:18013)
  • [Gr 1] D. Grayson, Products in $ K$-theory and intersecting algebraic cycles, Invent. Math. 47 (1978), 71-83. MR 0491685 (58:10890)
  • [Gr 2] -, Grothendieck rings and Witt vectors, Comm. Algebra 6 (1978), 249-255. MR 484183 (81e:13014)
  • [GQ] -, Higher algebraic $ K$-theory. II (after D. Quillen), Lecture Notes in Math., vol. 551, Springer-Verlag, Berlin, 1976. MR 0574096 (58:28137)
  • [Gro] A. Grothendieck, Crystals and the de Rham cohomology of schemes, Dix Exposés sur la Cohomologie des Schémas, North-Holland, Amsterdam, 1968. MR 0269663 (42:4558)
  • [HJ] C. Hood and J. Jones, Some algebraic properties of cyclic homology groups, $ K$-Theory 1 (1987), 361-384. MR 920950 (89f:18011)
  • [Ig] K. Igusa, What happens to Hatcher and Wagoner's formula for $ {\pi _0}C(M)$ when the first Postnikov invariant of $ M$ is nontrivial?, Lecture Notes in Math., vol. 1046, Springer-Verlag, Berlin,
  • [Ill] L. Illusie, Complexe de de Rham-Witt et cohomologie cristalline, Ann. Sci. École Norm. Sup. (4) 12 (1979), 501-661. MR 565469 (82d:14013)
  • [Ka] C. Kassel, Cyclic homology, comodules, and mixed complexes, J. Algebra 107 (1987), 195-216. MR 883882 (88k:18019)
  • [Laz] M. Lazard, Commutative formal groups, Lecture Notes in Math., vol. 443, Springer-Verlag, Berlin, 1975. MR 0393050 (52:13861)
  • [LQ] J.-L. Loday and D. Quillen, Cyclic homology and the Lie algebra homology of matrices, Comment. Math. Helv. 59 (1984), 565-591. MR 780077 (86i:17003)
  • [MS] H. Maazen and J. Steinstra, A presentation of $ {K_2}$ of split radical pairs, J. Pure Appl. Algebra 10 (1977), 271-294. MR 0472795 (57:12485)
  • [Q] D. Quillen, Higher algebraic $ K$-theory. I, Lecture Notes in Math., vol. 341, Springer-Verlag, Berlin, 1973. MR 0338129 (49:2895)
  • [St 1] J. Steinstra, Cartier-Dieudonne theory for Chow groups, J. Reine Angew. Math. 355 (1985), 1-66, and 362 (1985), 218-220. MR 809977 (88c:14026b)
  • [St 2] -, On $ {K_2}$ and $ {K_3}$ of truncated polynomial rings, Lecture Notes in Math., vol. 854, Springer-Verlag, Berlin, 1981. MR 618315 (82k:13016)
  • [Wa] J. Wagoner, Delooping classifying spaces in algebraic $ K$-theory, Topology 11 (1972), 349-370. MR 0354816 (50:7293)
  • [We 1] C. Weibel, Mayer-Vietoris sequences and module structures on $ N{K_*}$, Lecture Notes in Math., vol. 966, Springer-Verlag, Berlin, 1982. MR 618317 (82k:18010)
  • [We 2] -, $ K$-theory and analytic isomorphisms, Invent. Math. 61 (1980), 177-197. MR 590161 (83b:13011)
  • [We 3] -, Nil $ K$-theory maps to cyclic homology, Trans. Amer. Math. Soc. 303 (1987), 541-558. MR 902784 (89b:18014)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 19D55, 16E20, 18G50, 58A12

Retrieve articles in all journals with MSC: 19D55, 16E20, 18G50, 58A12


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-0978383-1
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society