Primary L-theory

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


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
$$L_h^n({S^{ - 1}}A[x],\varepsilon ) = L_h^n(A[x],\varepsilon ) \oplus LEnd_S^n(A,\varepsilon )$$
$$L_h^n({{\tilde S}^{ - 1}}A[x],\varepsilon ) = L_h^n(A[x],\varepsilon ) \oplus L\mathop \sim \limits_{End_s^n} (A,\varepsilon ),$$
$$LEnd_S^n(A,\varepsilon ) = L_p^n(A,\varepsilon ) \oplus L\mathop \sim \limits_{End_s^n} (A,\varepsilon )$$


Exact Sequence Integral Domain Module Chain Quadratic Case Splitting Theorem 
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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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