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Primary L-theory

  • Andrew Ranicki
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

The various types of L-groups defined in Chaps. 26,27,30,32 (asymmetric, automorphism, endomorphism, isometric etc.) have primary analogues in which the endomorphism f is required to be such that p(f) = 0 for some polynomial p(z) in a prescribed class of polynomials. The primary L-groups feature in the L-theory analogues of the splitting theorems of Chap. 12
$${K_1}({s^{ - 1}}A[x]) = {K_1}(A[x]) \oplus End_0^s(A)$$
$${K_1}({\tilde S^{ - 1}}A[x]) = {K_1}(A) \oplus \tilde {End}_0^S(A)$$
$$End_0^s(A) = {K_0}(A) \oplus \tilde {End}_0^S(A)$$
which will will now be obtained, with SA[x]a multiplicative subset with leading units and xS, and \(\tilde S \subset A[x]\) the reverse multiplicative subset (12.15). The S-primary endomorphism L-groups LEnd S * (A, є), \(L\tilde {Edn}_S^*\left( {A, \in } \right)\) are defined for an involution-invariant SA[x],such that
$$L_h^n({S^{ - 1}}A[x],\varepsilon ) = L_h^n(A[x],\varepsilon ) \oplus LEnd_S^n(A,\varepsilon )$$
$$L_h^n({{\tilde S}^{ - 1}}A[x],\varepsilon ) = L_h^n(A[x],\varepsilon ) \oplus L\mathop \sim \limits_{End_s^n} (A,\varepsilon ),$$
$$LEnd_S^n(A,\varepsilon ) = L_p^n(A,\varepsilon ) \oplus L\mathop \sim \limits_{End_s^n} (A,\varepsilon )$$
.

Keywords

Exact Sequence Integral Domain Module Chain Quadratic Case Splitting Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Andrew Ranicki
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of EdinburghEdinburghScotland, UK

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