Abstract
Given a ring R, it is known that the topological space BGl(R)+ is an infinite loop space. One way to construct an infinite loop structure is to consider the category F of free R-modules, or rather its classifying space BF, as food for suitable infinite loop space machines. These machines produce connective spectra whose zeroth space is (BF)+ = ZXBG1(R)+. In this paper we consider categories C o(F) = F, C 1(F),... of parametrized free modules and bounded homomorphisms and show that the spaces (BC o)+ = (BF)+, (BC 1)+,... are the connected components of a nonconnective ω-spectrum BC(F) with π iBC(F) = Ki(R) even for negative i.
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References
J.F. Adams, Infinite Loop Spaces, Princeton University Press, Princeton, 1978.
H. Bass, Algebraic K-theory, Benjamin, New York, 1968.
P. Freyd, Abelian Categories, Harper and Row, New York, 1964.
S.M. Gersten, On the spectrum of algebraic K-theory, Bull. AMS 78 (1972), 216–219.
D. Grayson, Higher algebraic K-theory: II (after D. Quillen), Lecture Notes in Math. No 551, Springer-Verlag, 1976.
M. Karoubi, La periodicite de Bott en K-theorie generale, Ann. Sci. Ec. Norm. Sup. (Paris) 4 (1971), 63–95.
M. Karoubi, Foncteur derives et K-theorie, Lecture Notes in Math. No. 136, Springer-Verlag, 1970.
M. Karoubi and O. Villamayor, K-Theorie algebrique et K-thorie topologique, C.R. Acad. Sci. (Paris) 269, serie A (1969), 416–419.
S. Maclane, Categories for the Working Mathematician, Springer-Verlag, 1971.
E.K. Pedersen, On the K−i functors, J. Algebra 90 (1984), 461–475.
D. Quillen, Higher algebraic K-theory: I, Lecture Notes in Math. No 341, Springer-Verlag, 1978.
R. Thomason, First quadrant spectral sequences in algebraic K-theory via homotopy colimits, Comm. in Alg. 10 (1982) 1589–1668.
J.B. Wagoner, On K2 of the Laurent polynomial ring, Amer. J. Math. 93 (1971), 123–138.
J.B. Wagoner, Delooping classifying spaces in algebnraic K-theory, Topology 11 (1972), 349–370.
D. Grayson, Localization for flat modules in Algebraic K-theory, J. of Algebra 61 (1979), 463–496.
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© 1985 Springer-Verlag
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Pedersen, E.K., Weibel, C.A. (1985). A nonconnective delooping of algebraic K-theory. In: Ranicki, A., Levitt, N., Quinn, F. (eds) Algebraic and Geometric Topology. Lecture Notes in Mathematics, vol 1126. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0074443
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DOI: https://doi.org/10.1007/BFb0074443
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