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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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$K$-theory of hyperplanes
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by Barry H. Dayton and Charles A. Weibel PDF
Trans. Amer. Math. Soc. 257 (1980), 119-141 Request permission

Abstract:

Let R be the coordinate ring of a union of N hyperplanes in general position in ${\textbf {A}}_k^{n + 1}$. Then \[ {K_i}(R) = {K_i}(k) \oplus \left ( {\begin {array}{*{20}{c}} {N - 1} \\ {n + 1} \\ \end {array} } \right ) {K_{n + i}}(k).\] 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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Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 257 (1980), 119-141
  • MSC: Primary 18F25; Secondary 14C35
  • DOI: https://doi.org/10.1090/S0002-9947-1980-0549158-6
  • MathSciNet review: 549158