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Transactions of the American Mathematical Society

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



Higher algebraic $ K$-theories

Authors: D. Anderson, M. Karoubi and J. Wagoner
Journal: Trans. Amer. Math. Soc. 226 (1977), 209-225
MSC: Primary 18F25; Secondary 16A54
MathSciNet review: 0444743
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Abstract: A homotopy fibration is established relating the Volodin or BN-pair definition of algebraic K-theory to the theory defined by Quillen.

In [2] we outlined the construction of natural homomorphisms

$\displaystyle K_ \ast ^Q \to K_ \ast ^{BN} \to K_ \ast ^V \to K_ \ast ^{KV}$

between higher algebraic K-theories $ K_ \ast ^Q$ of [10] and [11], $ K_ \ast ^{BN}$ of [17], $ K_ \ast ^V$ of [16], and $ K_ \ast ^{KV}$ of [7] and [8]. This was one of the steps in proving the various definitions of higher K-theory are equivalent. It turns out they all agree-including the theory $ K_ \ast ^S$ of [14], [5], and [8]-provided one restricts to the category of regular rings when using $ K_\ast^{KV}$. See [1], [2], [5], [8] and [18]. The purpose of this paper is to prove the following theorem, announced in [2], which yields the construction of $ K_ \ast ^Q \to K_ \ast ^{BN}$.

Theorem. For any associative ring with identity A

$\displaystyle G{L^{BN}}(A) \to B{\{ {U_F}\} ^ + } \to BGL{(A)^ + }$

is a homotopy fibration.

For the reader's convenience and because the presentation of the BN-pair K-theory $ K_ \ast ^{BN}$ used here is slightly different from that of [17], we shall briefly recall the definition of $ G{L^{BN}}$ and $ B\{ {U_F}\} $ in the first section.

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Keywords: Comparison of K-theories
Article copyright: © Copyright 1977 American Mathematical Society