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Transactions of the American Mathematical Society

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



$ K$-theory of hyperplanes

Authors: Barry H. Dayton and Charles A. Weibel
Journal: Trans. Amer. Math. Soc. 257 (1980), 119-141
MSC: Primary 18F25; Secondary 14C35
MathSciNet review: 549158
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Abstract: Let R be the coordinate ring of a union of N hyperplanes in general position in $ {\textbf{A}}_k^{n + 1}$. Then

$\displaystyle {K_i}(R)\, = \,{K_i}(k)\, \oplus \,\left( {\begin{array}{*{20}{c}} {N\, - \,1} \\ {n\, + \,1} \\ \end{array} } \right)\,{K_{n + i}}(k).$

This formula holds for $ {K_0},\,{K_1},\,{K_i}\,(i < 0)$, and for the Karoubi-Villamayor groups $ K{V_i}\,(i \in \,{\mathbf{Z}})$. For $ {K_2}$ there is an extra summand $ \bar R/R$, where $ \bar R$ is the normalization of R. For $ {K_3}$ the above is a quotient of $ {K_3}(R)$.

In §4 we show that $ {K_1}$-regularity implies $ {K_0}$-regularity, answering a question of Bass. We also show that $ {K_i}$-regularity is equivalent to Laurent $ {K_i}$-regularity for $ i \leqslant 1$. The results of this section are independent of the rest of the paper.

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Keywords: Karoubi-Villamayor K-theory, $ {K_i}$-regular, scheme, geometric realization, poset, Tate map
Article copyright: © Copyright 1980 American Mathematical Society