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

Ryan, John G.

doi:10.1090/S0002-9947-1990-0978383-1

Curves on $K$-theory and the de Rham homology of associative algebras
HTML articles powered by AMS MathViewer

by John G. Ryan PDF

Trans. Amer. Math. Soc. 321 (1990), 559-582 Request permission

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

Spencer Bloch, Algebraic $K$-theory and crystalline cohomology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 187–268 (1978). MR 488288
Nicolas Bourbaki, Éléments de mathématique, Masson, Paris, 1980 (French). Algèbre. Chapitre 10. Algèbre homologique. [Algebra. Chapter 10. Homological algebra]. MR 610795
Dan Burghelea and Crichton Ogle, The Künneth formula in cyclic homology, Math. Z. 193 (1986), no. 4, 527–536. MR 867346, DOI 10.1007/BF01160473
S. Geller, L. Reid, and C. Weibel, The cyclic homology and $K$-theory of curves, J. Reine Angew. Math. 393 (1989), 39–90. MR 972360, DOI 10.1515/crll.1989.393.39
Thomas G. Goodwillie, Relative algebraic $K$-theory and cyclic homology, Ann. of Math. (2) 124 (1986), no. 2, 347–402. MR 855300, DOI 10.2307/1971283
Thomas G. Goodwillie, On the general linear group and Hochschild homology, Ann. of Math. (2) 121 (1985), no. 2, 383–407. MR 786354, DOI 10.2307/1971179
Daniel R. Grayson, Products in $K$-theory and intersecting algebraic cycles, Invent. Math. 47 (1978), no. 1, 71–83. MR 491685, DOI 10.1007/BF01609480
Daniel R. Grayson, Grothendieck rings and Witt vectors, Comm. Algebra 6 (1978), no. 3, 249–255. MR 484183, DOI 10.1080/00927877808822245
Daniel Grayson, Higher algebraic $K$-theory. II (after Daniel Quillen), Algebraic $K$-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976) Lecture Notes in Math., Vol. 551, Springer, Berlin, 1976, pp. 217–240. MR 0574096
A. Grothendieck, Crystals and the de Rham cohomology of schemes, Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math., vol. 3, North-Holland, Amsterdam, 1968, pp. 306–358. Notes by I. Coates and O. Jussila. MR 269663
Christine E. Hood and John D. S. Jones, Some algebraic properties of cyclic homology groups, $K$-Theory 1 (1987), no. 4, 361–384. MR 920950, DOI 10.1007/BF00539623

What happens to Hatcher and Wagoner’s formula for

when the first Postnikov invariant of

is nontrivial

Luc Illusie, Complexe de de Rham-Witt et cohomologie cristalline, Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 4, 501–661 (French). MR 565469, DOI 10.24033/asens.1374
Christian Kassel, Cyclic homology, comodules, and mixed complexes, J. Algebra 107 (1987), no. 1, 195–216. MR 883882, DOI 10.1016/0021-8693(87)90086-X
Michel Lazard, Commutative formal groups, Lecture Notes in Mathematics, Vol. 443, Springer-Verlag, Berlin-New York, 1975. MR 0393050, DOI 10.1007/BFb0070554
Jean-Louis Loday and Daniel Quillen, Cyclic homology and the Lie algebra homology of matrices, Comment. Math. Helv. 59 (1984), no. 4, 569–591. MR 780077, DOI 10.1007/BF02566367
Hendrik Maazen and Jan Stienstra, A presentation for $K_{2}$ of split radical pairs, J. Pure Appl. Algebra 10 (1977/78), no. 3, 271–294. MR 472795, DOI 10.1016/0022-4049(77)90007-X
Daniel Quillen, Higher algebraic $K$-theory. I, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 85–147. MR 0338129
Jan Stienstra, Cartier-Dieudonné theory for Chow groups, J. Reine Angew. Math. 355 (1985), 1–66. MR 772483, DOI 10.1515/crll.1985.355.1
Jan Stienstra, On $K_{2}$ and $K_{3}$ of truncated polynomial rings, Algebraic $K$-theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980) Lecture Notes in Math., vol. 854, Springer, Berlin, 1981, pp. 409–455. MR 618315, DOI 10.1007/BFb0089532
J. B. Wagoner, Delooping classifying spaces in algebraic $K$-theory, Topology 11 (1972), 349–370. MR 354816, DOI 10.1016/0040-9383(72)90031-6
C. A. Weibel, Mayer-Vietoris sequences and module structures on $NK_\ast$, Algebraic $K$-theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980) Lecture Notes in Math., vol. 854, Springer, Berlin, 1981, pp. 466–493. MR 618317, DOI 10.1007/BFb0089534
Charles A. Weibel, $K$-theory and analytic isomorphisms, Invent. Math. 61 (1980), no. 2, 177–197. MR 590161, DOI 10.1007/BF01390120
Charles A. Weibel, Nil $K$-theory maps to cyclic homology, Trans. Amer. Math. Soc. 303 (1987), no. 2, 541–558. MR 902784, DOI 10.1090/S0002-9947-1987-0902784-0