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Algebraic $ K$-theory via binary complexes


Author: Daniel R. Grayson
Journal: J. Amer. Math. Soc. 25 (2012), 1149-1167
MSC (2010): Primary 19D99
DOI: https://doi.org/10.1090/S0894-0347-2012-00743-7
Published electronically: June 14, 2012
MathSciNet review: 2947948
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Abstract: Motivated by work of Nenashev on $ K_1$, we introduce acyclic binary multicomplexes and use them to provide generators and relations for the Quillen $ K$-groups of an arbitrary exact category.


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  • 1. A. K. Bousfield and E. M. Friedlander, Homotopy theory of $ \Gamma $-spaces, spectra, and bisimplicial sets, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, Lecture Notes in Math., vol. 658, Springer, Berlin, 1978, pp. 80-130. MR 513569 (80e:55021)
  • 2. Denis-Charles Cisinski, Théorèmes de cofinalité en $ k$-théorie, d'après Thomason, available at http://www.math.univ-toulouse.fr/~dcisinsk/cofdev.dvi, December 2002.
  • 3. Daniel R. Grayson, Localization for flat modules in algebraic $ K$-theory, J. Algebra 61 (1979), no. 2, 463-496. MR 559850 (82c:14011)
  • 4. -, Exact sequences in algebraic $ K$-theory, Illinois J. Math. 31 (1987), no. 4, 598-617. MR 909785 (89c:18011)
  • 5. -, Adams operations on higher $ K$-theory, $ K$-Theory 6 (1992), no. 2, 97-111. MR 1187703 (94g:19005)
  • 6. -, Weight filtrations via commuting automorphisms, $ K$-Theory 9 (1995), no. 2, 139-172. MR 1340843 (96h:19001)
  • 7. John Milnor, Introduction to algebraic $ K$-theory, Princeton University Press, Princeton, NJ, 1971, Annals of Mathematics Studies, No. 72. MR 0349811 (50:2304)
  • 8. Alexander Nenashev, Double short exact sequences produce all elements of Quillen's $ K_1$, Algebraic $ K$-theory (Poznań, 1995), Contemp. Math., vol. 199, Amer. Math. Soc., Providence, RI, 1996, pp. 151-160. MR 1409623 (97g:19001)
  • 9. -, Double short exact sequences and $ K_1$ of an exact category, $ K$-Theory 14 (1998), no. 1, 23-41. MR 1621690 (99e:19001)
  • 10. -, $ K\sb 1$ by generators and relations, J. Pure Appl. Algebra 131 (1998), no. 2, 195-212. MR 1637539 (99g:19001)
  • 11. 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 (49:2895)
  • 12. Marco Schlichting, Higher algebraic $ K$-theory, Topics in algebraic and topological $ K$-theory, Lecture Notes in Math., vol. 2008, Springer, Berlin, 2011, pp. 167-241. MR 2762556 (2012a:19001)
  • 13. Clayton Sherman, On $ K_1$ of an abelian category, J. Algebra 163 (1994), no. 2, 568-582. MR 1262719 (95a:19001)
  • 14. -, On $ K_1$ of an exact category, $ K$-Theory 14 (1998), no. 1, 1-22. MR 1621689 (99e:19002)
  • 15. -, $ K_1$ of Exact Categories by Mirror Image Sequences, J. $ K$-Theory (2012), available online, May, 2012.
  • 16. Michael R. Stein and R. Keith Dennis, $ K_{2}$ of radical ideals and semi-local rings revisited, Algebraic $ K$-theory, II: ``Classical'' algebraic $ K$-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Math., Vol. 342, Springer, Berlin, 1973, pp. 281-303. MR 0406998 (53:10782)
  • 17. R. W. Thomason and Thomas Trobaugh, Higher algebraic $ K$-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol.III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247-435. MR 1106918 (92f:19001)
  • 18. Friedhelm Waldhausen, Algebraic $ K$-theory of spaces, Algebraic and geometric topology (New Brunswick, N.J., 1983), Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 318-419. MR 802796 (86m:18011)

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Additional Information

Daniel R. Grayson
Email: drg@illinois.edu

DOI: https://doi.org/10.1090/S0894-0347-2012-00743-7
Received by editor(s): October 21, 2011
Received by editor(s) in revised form: May 11, 2012
Published electronically: June 14, 2012
Additional Notes: The author was supported by the National Science Foundation under grants NSF DMS 08-10948 and NSF DMS 10-02171
Dedicated: This paper is dedicated to the memory of Daniel Quillen.
Article copyright: © Copyright 2012 Daniel Grayson; dedicated 2022 to the public domain

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